My two standard proof-texts for arguing that the promises of the OT find their fulfillment in Christ are:
2 Corinthians 1:20: For no matter how many promises God has made, they are "Yes" in Christ. And so through him the "Amen" is spoken by us to the glory of God.
Acts 13:32-33: "We tell you the good news: What God promised our fathers he has fulfilled for us, their children, by raising up Jesus.
I would add that the promises referred to in both passages are specifically the Abrahamic promises of blessings for the nations, land, and descendents.
