For those who would deny that the Septuagint even exists (as I read once in a KJV-only journal), I would point to the following:
1. We have LXX manuscripts that predate the Incarnation and the time of the Apostles (e.g., the Rylands 458 papyrus on Deuteronomy, dated to the 2nd century BC) 2. Plenty of passages exist in the Greek NT (whatever text-type you prefer) that actually agrees with LXX manuscripts against the Masoretic (e.g., Romans 15:12; in addition, simply go through the book of Hebrews, which is predominantly LXX, including both where it is similar to the Hebrew MT and where it isn't). This rules out the possibility that the Apostles were simply creating their own translation on the fly (at least in some cases).
End introductory note.
To the latter end, I threw a "competency" exam at them the last day of class where they had to translate from a chunk of the Didache (approx. 100-130 AD, give or take a few decades). However, throughout the semester, I gave them "translation and parsing" quizzes taken straight from the canonical Septuagint (by "Septuagint," a.k.a. LXX, a.k.a. "Old Greek," I mean the version of the Old Testament translated in Alexandria over a period of many decades before the birth of Jesus. There were other Greek translations that came later).
The Septuagint is, to a significant degree, the "Bible of the early church." The average Gentile Christian (and perhaps many Hellenized Jewish Christians) would not have been able to read Hebrew. Consequently, the LXX was used by the apostles quite often (though not always) when quoting the Old Testament. Consequently, the LXX is a key part of New Testament studies. By default, the LXX usage of a particular word should be considered before (though not in exclusion of) the usage by Josephus, Philo, or secular Greek authors. Case in point: "hilasterion" in Romans 3:25 (the word only occurs twice in the NT) is used in the LXX Torah to refer to the physical Mercy Seat in the tabernacle (quite frequently!) This would obviously be in the back of the mind of the majority of Greek-speaking Christians reading Romans 3:25, at least Christians with even a cursory familiarity with the Old Testament). In other words, the LXX is in one sense the "lexical currency" of NT writers (some more than others).
How the LXX should be viewed regarding inspiration, canonicity, etc. is an open question within certain areas of Christian scholarship (which is why Augustine had his infamous debate with Jerome over the latter's new translation). Yet the most that can be said is that the NT writers viewed a translation as an adequate vehicle for conveying God's word, even if it differed somewhat from the originals.
This is hardly the tip of the iceberg, of course! LXX research is an exciting, ongoing field of study (for what it's worth, I recently had an article accepted for publication that suggests Peter "corrected" the LXX in 1 Peter 2:6 for various reasons). For the reader who's interested in this topic, I would recommend as the essential starting point the book Invitation to the Septuagint by Karen H. Jobes and Moises Silva.