There are several helpful articles in volume one of The Expositor’s Bible Commentary (ed. Frank Gaebelein). One of them that stands out is Roger Nicole’s “The Old Testament in the New Testament.” Here is Nicole’s introduction and his main points (edited and summarized).
“One very notable feature of the NT is the extent to which it alludes to or quotes the OT. It appeals to the OT in order to provide proof of statements made, confirmation for positions espoused, illustration of principles advanced, and answers to questions raised. …[There is] a very close relationship between the Testaments. Eight propositions clarify this relationship.”
1) The NT writers assumed that the OT in its entirety was meaningful and relevant for their own time.
2) The NT writers were convinced that many of the events of the life of our Lord and indeed of the beginnings of the Christian church had been prophesied in considerable detail by OT writers.
3) The ground of the NT writers’ faith in the prophetic vision of the OT was their conviction, frequently and variously expressed, that the OT is the Word of God.
4) Because they viewed the OT as the Word of God, the NT writers did not hesitate to interpret its statements, not merely in terms of what the human authors could have thought, but in terms of what God himself meant in speaking through the prophets.
5) In many cases the NT writers, illumined by the Holy Spirit, perceived with greater clarity than the OT writers themselves God’s intended meaning behind some prophecies. What the prophets had seen only dimly and in terms of general principle, the NT writers saw in the glowing light of fulfillment in a perspective in which a wealth of details fall into place.
6) The NT writers had such a deep insight into the fullness of God’s redemptive purposes that they could perceive foreshadowings and parallelisms where others might easily have missed them altogether. In many such cases it is not necessary to hold that the OT writers completely understood the way their pronouncements would relate to their fulfillment in the NT.
7) In a number of cases the NT authors saw a significant relationship between a diversity of OT passages. Sometimes they made this plain by a juxtaposition of quotations; in other cases, they appear to have united two or more passages in an illuminating combination.
8) While the NT writers draw attention mainly to the meaning of OT passages, they did not hesitate to build an argument on one word of the original text. This method of quoting the OT manifests a supreme confidence in the divine authority of even then minutest details of Scripture.
I appreciate these 8 points; Nicole does give a lot of biblical support and examples for each of these points. I recommend reading the entire article for a stimulating study of how the NT authors used the OT. Near the end, Nicole says this (and I’ll end with it):
“Our acceptance of the Bible doctrine of inspiration carries with it the assurance that the Holy Spirit enlightened and guided the writers of the NT in their understanding of the OT no less than he enlightened and guided the OT writers in whatever they wrote.”