Davis drained all eight of his free-throw tries for seventh-seeded Fresno State (17-15), which will play No. 2 seed New Mexico in the quarterfinals on Thursday. Tyler Johnson paced the winning effort with 19 points and 12 rebounds, while Allen Huddleston added 15 points to the mix.
For 10th-seeded Air Force (12-18), Tre' Coggins led all scorers with 20 points and Max Yon netted 15.
Fresno State carried a 27-24 lead into halftime even though it shot a dreadful 27.6 percent from the field in the first half. It went an impressive 9-of-11 from the foul line and had five more field goal attempts than Air Force, which also had a poor shooting performance in the opening 20 minutes (.333).
The Falcons stormed back by piecing together a 26-10 run in the second half, and they held a seemingly comfortable 56-41 lead with under six minutes remaining. However, Fresno State ended the game on a furious 20-3 run. After a Johnson 3-pointer tied the game with 1:16 left, the teams exchanged empty possessions before Davis had a 2-of-2 trip to the stripe with 12 seconds remaining. Air Force had a chance to tie on its final possession but a Coggins jumper in the waning seconds did not fall.
Fresno State scored 13 points off of 12 Air Force turnovers.