The Gospel Coalition, a Calvinist site, writes about the New Testament use of the Old Testament:
1. Keep in mind the NT’s purpose in referencing the OT. We often think every time the OT is referenced it must mean the NT author is trying to exegete the OT passage. But there is no rule of inerrancy which says the NT author must always be attempting to give the correct interpretation of a given passage. The NT author may not be attempting an interpretation at all. If someone asks me, “How is the editing work going” and I say, “It’s tedious–line upon line, precept upon precept” this doesn’t mean I’m trying to exegete Isaiah 28:10. I’m simply employing the familiar language of a familiar passage.
2. Along these lines, remember the NT often uses the OT simply as a vehicle of expression. The NT writers were hugely familiar with the OT. It’s no wonder they employed its vocabulary. In the same way, Westerners might use a line from Shakespeare or the Bible because it is familiar, but without intending to explain its context or original meaning.
3. The NT may press home the significance of a passage without trying to explain its original meaning. For example, Moo points to Paul’s use of Deuteronomy 25:4 (“You shall not muzzle an ox when it treads out the grain”) in 1 Corinthians 9:9. Critics argue that Paul is taking the Law of Moses out of context by saying this passage is about paying ministers. But surely Paul is justified in pulling a fair inference out of the passage and applying it to his own context.
4. We must allow for a broader view of “fulfillment” language. A lot of trouble could be avoided if we understood that the use of plēroō (fulfilled) does not have to mean “and so this verse predicted that Jesus would do or say this thing that just happened.” As Moo says, “The word is used in the New Testament to indicate the broad redemptive-historical relationship of the new, climactic revelation of God in Christ to the preparatory, incomplete revelation to and through Israel” (191). In other words, “fulfilled” does not mean the OT text in question is a direct prophecy. Consequently, Jesus flight to Egypt can fulfill Hosea 11:1, not because Hosea ever intended to predict a Messianic trip down south, but because Jesus is God’s greater Son who is the embodiment of a new Israel. Jesus is on an Exodus journey of his own. Hosea did not predict the Holy Family’s flight to Egypt, nor does Matthew suggest the prophet meant to do so. But Matthew does see that the story of Israel’s exodus, alluded to in Hosea, is brought to its full redemptive-historical revelation in Christ.
5. Similarly, some OT passages are fulfilled typologically. This is different than allegory. And allegory looks for meaning behind the text where typology finds a developed meaning that is rooted in the text (see Moo 195). Jesus’ passion can be seen as a fulfillment of David’s heart cry in Psalm 22 not because David thought he was predicting the death of the Messiah, but because David, as the king and as the promised progenitor of the Messiah, was a type of Christ whose cries anticipated the final dereliction of David’s greater son.
6. OT prophecy is full of examples where there is a near and far fulfillment. Isaiah 40, for example, was a word of comfort about the return from Babylon, but later we see it also was a word about John the Baptist who would prepare the way for the Messiah (Mark 1:2-3). Much of the prophetic witness implicitly anticipates a future, fuller, often eschatological fulfillment. Isaiah may not have known that his words about the virgin were Messianic, but this does not mean he’d be surprised to know they were. Israel was always waiting for the everlasting kingdom and the final Deliverer. I think the prophets understood that what they foretold (and forth-told) was for their day, but it could be for the future as well.