This is how St. John Chrysostom explains it (On Virginity pp. 78-79 // PDF pp. 94-95):

3. And so Paul said: "Not the Lord, but I," [1 Cor. 7:12] not because he wished to indicate that these words were of human origin—how could they be?—but that the Lord had given this command not when he was present with his disciples, but now, through Paul. Accordingly, just as the phrase, "The Lord, not I," [1 Cor. 7:10] does not contradict the command of Christ, so the phrase, "I, not the Lord," [1 Cor. 7:12] is in no way a personal expres­sion contrary to God's will but an indication of this only, that his command is now being given through his apostle.

I am starting to believe differently. 1 cor 9 shows that there were those that didnt feel Paul was an Apostle. He made efforts to show he was. But I think people have begun making Paul's words out to be a new Mosaic Law. I believe that we are disciplining the congregation, as if Paul's words are matters of left or right, write or wrong. Jesus was the one that saved and gave his life for us, not Paul. Paul openly says not to place him on high in this regard, all through 1 corinthians. It seems like we have become the sort that assumes any and all that Paul said cant be from his personal view, even when he openly says it is. Can't he pass a view on a matter and it just be a suggestion and not a law? Have we elevated Paul into a sect? Paul once said, "I follow Paul, I follow Apollos, I follow Jesus." Are we now Paul followers, even trumping Jesus' expectations? Are we following Paul's words as greater than his personal views?

Paul gives us a clean incite into the way Christianity was getting organized (ie those taking the lead, and how, and whom)But was he building a law? Jesus said for example, "Stop judging your brother" Paul however said "we judge those on the inside, but God Judges those on the outside"

Listen to John, Who said he had a Revelation given him by Jesus: Nevertheless, I have a few things against you: There are some among you who hold to the teaching of Balaam, who taught Balak to entice the Israelites to sin so that they ate food sacrificed to idols and committed sexual immorality.---rev 2:14

then listen to the original Apostles: to keep abstaining from things sacrificed to idols, from blood, from what is strangled, and from sexual immorality.-Acts 15:29

Listen to John Who had a revelation From Jesus, in 64 CE :Nevertheless, I have a few things against you: There are some among you who hold to the teaching of Balaam, who taught Balak to entice the Israelites to sin so that they ate food sacrificed to idols and committed sexual immorality.--REV 2:14

However, Listen to the logic of Paul... just as there are many “gods” and many “lords,”6 there is actually to us one God, the Father, from whom all things are and we for him; and there is one Lord, Jesus Christ, through whom all things are and we through him.7 However, not all have this knowledge. But some, because of their former association with the idol, eat food as something sacrificed to an idol, and their conscience, being weak, is defiled.8 But food will not bring us nearer to God; we are no worse off if we do not eat, nor better off if we eat.9 But keep watching that your right to choose does not somehow become a stumbling block to those who are weak.10 For if anyone should see you who have knowledge having a meal in an idol temple, will not the conscience of that one who is weak be emboldened to the point of eating food offered to idols?11 So by your knowledge the man who is weak is being ruined, your brother for whose sake Christ died.12 When you sin against your brothers in this way and wound their weak conscience, you are sinning against Christ.-----1st cor 8:6-12

If an unbeliever invites you and you want to go, eat whatever is set before you, making no inquiry on account of your conscience.28 But if anyone says to you, “This is something offered in sacrifice,” do not eat because of the one who told you and because of conscience.29 I do not mean your own conscience, but that of the other person. For why should my freedom be judged by another person’s conscience?30 If I am partaking with thanks, why am I to be spoken of abusively over that for which I give thanks?31 Therefore, whether you are eating or drinking or doing anything else, do all things for God’s glory.32 Keep from becoming causes for stumbling to Jews as well as Greeks and to the congregation of God,33 just as I am trying to please all people in all things, not seeking my own advantage, but that of the many, so that they may be saved.---1cor 10:27-32

So wait, when John was saying "watch out for the one that gets us to eat food sacrificed to idols, written 10 years after paul, was he referencing the teachings of Paul?

Evangelicals covers a wide group with differing beliefs. However let me try to address this from a widely held Christian perspective.

Most Christians do not find a problem with this line. Problems that are raised are often because of a distorted understanding of the nature of scripture.

Unlike other religion's beliefs about their scriptures, Christians do not believe that the Bible is literally the dictated word of God. Nor is it simply a set of commands, direct from the mouth of God, applicable to all Christians in all times. Instead Christians believe that God used the human authors, including their own styles and intellects, to write scriptures that would convey his message. Christians believe the Bible to be correct, but not that it is universally applicable.

Taken like this there is nothing the matter with Paul including his own wisdom within a scriptural letter. It is correct (correct in the sense that the letter truly reports what Paul thinks about this) and so does not in any way invalidate the infallibility of scripture, but it identifies that the statement is not a direct command from God. We may wonder why God caused Paul to write this, but it doesn't invalidate the passage, the letter, or the Bible.

Evangelical commentators generally hold that Paul is speaking authoritatively in 1 Corinthians 7:12, and only indicating that he was not aware of any direct teaching on this subject by Jesus during his earthly ministry. The Reformation Study Bible explains:

With regard to the problem treated in vv. 10, 11, there was a well-known instruction given by Jesus during His earthly ministry (Mark 10:1–12). In vv. 12–16, however, Paul discusses a difficult situation that had not been expressly addressed by the Lord while He walked the earth. His apostolic commands, however, come by inspiration and have divine authority, as 14:37 makes clear.

Others who make this point include The New Bible Commentary:

The problem of mixed marriages would hardly have arisen during Christ's ministry, and so Paul cannot quote a specific word of the Lord. But his own judgment is given authoritatively.

And Barry G. Webb, writing in The Enduring Authority of the Christian Scriptures, edited by D. A. Carson, says:

Paul's distinction between what he gives as a command of Christ ("not I, but the Lord") and what he gives only as his own word ("I, not the Lord") cannot be taken as distinguishing between what has apostolic authority and what does not. He writes the entire epistle as "an apostle of Christ Jesus by the will of God" (1:1). The basic distinction seems to be between appealing to a specific teaching of Jesus, and speaking as an apostle without being able to appear to any specific teaching of Jesus regarding the issue at hand. (chapter 19, fn. 17)

The ESV Study Bible says:

Paul views his admonition here as authoritative and inspired, not merely as human wisdom (v. 40; cf. 14:37–38).

The MacArthur Study Bible:

Not a denial of inspiration or an indication that Paul is giving human opinion, but simply a way of saying that Jesus had not spoken on this and God had not previously given revelation on the matter, as Paul was then writing.

Constable's Commentary agrees:

For mixed couples, Paul could not cite a teaching of Jesus, because He had not spoken on this subject. At least, as far as Paul knew He had not. Nevertheless, the risen Lord inspired Paul's instructions on this subject, so they were every bit as authoritative as the teaching Jesus gave during His earthly ministry.

And so does the Moody Bible Commentary:

Paul was not aware of any teaching on mixed marriages given by Jesus, but as an apostle inspired by God, Paul's words are binding.

Conclusion

Evangelicals broadly agree that what Paul writes in 7:12 is inspired and just as authoritative as the other teachings that Paul gives.

That said, it's perhaps worth noting that Evangelicals accept that some apostolic speeches and writings are not strictly inerrant in their content (though the record is still accurate). For example, they admit that Paul speaks erroneously in Acts 20:25, and that in 2 Corinthians 8, he gives advice, not a divine command. On the latter case, Barry Webb writes:

Just as the Old Testament Scriptures contain "command" (law) and "advice" (wisdom teaching), so do the writings of the apostles; but all of it is Scripture.

But with respect to the verses in question here, these comments are merely tangential – Evangelicals generally accept 1 Corinthians 7:10ff. as authoritative commands, under the inspiration of God.