The Oxford Companion to the Bible structures Ephesians in three parts: 

1. God's great work is completed through Jesus Christ and his Spirit (chapters 1-2)
2. The publication and continuation of God's work (3.1-4.16).
3. The testimony of daily life.

Chapter three is summarized in the following way:

To Paul and other apostles before him it was revealed that Jews and gentiles are common heirs of God's fulfilled promises. The suffering of the apostle in prison, for the sake of this message, is reason for joy, not despair. (b)3.14-21: In his prayer, Paul asks the Father for Christ's presence in every heart, and for increased appreciation of the love bestowed through Christ. 186

This two-part structure is generally understood by most readers.

Transitionally, the writer of Ephesians  links the beginning of chapter three back to the arguments of chapter two: "Eph 3:1 For this cause I Paul, the prisoner of Christ Jesus in behalf of you Gentiles,--" What is this cause? It is that Jews and gentiles might be reconciled: "Eph 2:16 and might reconcile them both in one body unto God through the cross."  This unifying of God's universal people, the writer refers to as the mystery made known by revelation:

Eph 3:2 if so be that ye have heard of the dispensation of that grace of God which was given me to you-ward;

Eph 3:3 how that by revelation was made known unto me the mystery, as I wrote before in few words,

Eph 3:4 whereby, when ye read, ye can perceive my understanding in the mystery of Christ;

Eph 3:5 which in other generation was not made known unto the sons of men, as it hath now been revealed unto his holy apostles and prophets in the Spirit;

Eph 3:6 to wit, that the Gentiles are fellow-heirs, and fellow-members of the body, and fellow-partakers of the promise in Christ Jesus through the gospel,

The dispensation of the grace of God, the writer says, is give "you-ward" but "in other generation was not made known unto the sons of men, as it hath now been revealed unto his holy apostles and prophets in the Spirit." And what is that mystery: "that the Gentiles are fellow-heirs, and fellow-members of the body, and fellow-partakers of the promise in Christ Jesus through the Gospel.

The writer of Ephesians next acknowledges that by revelation the responsibility and mission for making the mystery of Christ known to the gentiles is the author's:

Eph 3:7 whereof I was made a minister, according to the gift of that grace of God which was given me according to the working of his power.

Eph 3:8 Unto me, who am less than the least of all saints, was this grace given, to preach unto the Gentiles the unsearchable riches of Christ;

Eph 3:9 and to make all men see what is the dispensation of the mystery which for ages hath been hid in God who created all things;

Eph 3:10 to the intent that now unto the principalities and the powers in the heavenly places might be made known through the church the manifold wisdom of God,

Eph 3:11 according to the eternal purpose which he purposed in Christ Jesus our Lord:

Eph 3:12 in whom we have boldness and access in confidence through our faith in him.

If the author is Paul, as many have good reason for accepting, then his is a ministry in grace, and he is less than the least of all the saints. If someone other than Paul wrote this, one would wonder why the writer humbled Paul in this way. God, creator of all things,  has made the writer the minister of the dispensation of the mystery hidden for ages to the end that "the principalities and the powers in the heavenly places might be made known through the church the manifold wisdom of God." God's purpose is one and eternal: "purposed in Christ Jesus our Lord," the mediator through whom we (Jew and gentile) have access through faith. The writer explains that any tribulation associated with the mission has been willingly undertaken, that the writer has willingly subordinated himself to the Father for "whom every family in heaven and on earth is named":

Eph 3:13 Wherefore I ask that ye may not faint at my tribulations for you, which are your glory.

Eph 3:14 For this cause I bow my knees unto the Father,

Eph 3:15 from whom every family in heaven and on earth is named.

 

The last section of Ephesians 3 is a prayer:

Eph 3:16 that he would grant you, according to the riches of his glory, that ye may be strengthened with power through his Spirit in the inward man;

Eph 3:17 that Christ may dwell in your hearts through faith; to the end that ye, being rooted and grounded in love,

Eph 3:18 may be strong to apprehend with all the saints what is the breadth and length and height and depth,

Eph 3:19 and to know the love of Christ which passeth knowledge, that ye may be filled unto all the fulness of God.

Eph 3:20 Now unto him that is able to do exceeding abundantly above all that we ask or think, according to the power that worketh in us,

Eph 3:21 unto him be the glory in the church and in Christ Jesus unto all generations for ever and ever. Amen.

The prayer is that God through his Spirit  would strengthen the saints of Ephesus "in the inward man" in order that Christ might dwell in their hearts through faith; that they be rooted and grounded in love, apprehending the length, width, height, and depth and love of Christ that passeth knowledge; that they be filled with the fullness of God. The prayer concludes by acknowledge that God is "able to do exceeding abundantly above all that we ask or think." Unto God, then, "be the glory in the church and in Christ Jesus unto all generations for ever and ever. Amen."

Daniel Wallace has outlined chapter three of Ephesians in the following way:

This is the same structure I have just describes: Paul's relation to the mystery revealed and a prayer for love to maintain the church's practical unity. http://www.bible.org/docs/soapbox/epohl.htm

The reader may want to note the similarity of Ephesians three to Colossians three:

 

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For it pleased the Father that in him should all fulness dwell;

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And, having made peace through the blood of his cross, by him to reconcile all things unto himself; by him, I say, whether they be things in earth, or things in heaven.

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And you, that were sometime alienated and enemies in your mind by wicked works, yet now hath he reconciled

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In the body of his flesh through death, to present you holy and unblameable and unreproveable in his sight:

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If ye continue in the faith grounded and settled, and be not moved away from the hope of the gospel, which ye have heard, and which was preached to every creature which is under heaven; whereof I Paul am made a minister;

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Who now rejoice in my sufferings for you, and fill up that which is behind of the afflictions of Christ in my flesh for his body's sake, which is the church:

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Whereof I am made a minister, according to the dispensation of God which is given to me for you, to fulfil the word of God;

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Even the mystery which hath been hid from ages and from generations, but now is made manifest to his saints:

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To whom God would make known what is the riches of the glory o

f this mystery among the Gentiles; which is Christ in you, the hope of glory:

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Whom we preach, warning every man, and teaching every man in all wisdom; that we may present every man perfect in Christ Jesus:

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Whereunto I also labour, striving according to his working, which worketh in me mightily.

Here, too, the reconciliation of "all things" is by Christ, this time, not just Jews and gentiles but "things in heaven" and "things in earth." The writer is made a minister, "according to the dispensation of God which is given to me for you, to fulfill the word of God." Again, the mystery, "hid for ages and from generations," is now made manifest to the saints and among the gentiles.

The Net Bible also divides Ephesians three into two main parts: the writer's relation to divine mystery and a prayer for strengthened love:

Paul's Relationship to the Divine Mystery

3:1For this reason I, Paul, the prisoner of Christ Jesus for the sake of you Gentiles-- 3:2 If indeed1 you have heard of the stewardship2 of God's grace that was given to me for you, 3:3 that3 by revelation the divine secret4 was made known to me, as I wrote before briefly.5 3:4 When reading this,6 you will be able to7 understand my insight into this secret8 of Christ. 3:5 Now this secret9 was not disclosed to mankind10 in former11 generations as it has now been revealed to his holy apostles and prophets by12 the Spirit, 3:6 namely, that through the gospel13 the Gentiles are fellow heirs, fellow members14 of the body, and fellow partakers of the promise in Christ Jesus. 3:7 I became a servant of this gospel15 according to the gift of God's grace that was given to me by16 the exercise of his power.17 3:8 To me--less than the least of all the saints18--this grace was given,19 to proclaim to the Gentiles the unfathomable riches of Christ 3:9 and to enlighten20 everyone about the divine secret's plan21--a secret that has been hidden for ages22 in the God who has created all things. 3:10 The purpose of this enlightenment is that23 through the church the multifaceted wisdom24 of God should now be disclosed to the rulers and the authorities in the heavenly realms. 3:11 This was according to25 the eternal purpose that he accomplished in Christ Jesus our Lord, 3:12 in whom we have boldness and confident access26 to God27 because of28 Christ's29 faithfulness.30 3:13 For this reason I ask you31 not to lose heart because of what I am suffering for you,32 which33 is your glory.34

Prayer for Strengthened Love

3:14 For this reason35 I bow my knees before the Father, 3:15 from36 whom every family37 in heaven and on the earth is named. 3:16 I pray that38 according to the wealth of his glory he may grant you to be strengthened with power through his Spirit in the inner man, 3:17 that Christ may dwell in your hearts through faith, so that, by being rooted and grounded in love, 3:18 you may be able to comprehend with all the saints what is the breadth and length and height and depth,39 3:19 and thus to know the love of Christ that surpasses knowledge, so that you may be filled up to40 all the fullness of God.

3:20 Now to him who by the power that is working within us41 is able to do far beyond42 all that we ask or think, 3:21 to him be the glory in the church and in Christ Jesus to all generations, forever and ever. Amen.