Lee Cummard contributed 12 points and 10 rebounds for BYU (17-5, 5-3 Mountain West), while Chris Miles had 10. Tavernari became the 39th member of BYU's 1,000-point club with two free throws early in the second half.
The Cougars squandered a 13-point lead, falling behind 45-43 with 10 minutes, 14 seconds remaining, but went on a 26-3 run to secure their third win in four games.
Air Force (9-12, 0-8) was infused with energy thanks to a cache of freshmen, three of whom were making their first starts. The five freshmen combined for 21 points as Sammy Schafer led the way with a career-high 12 points.
Matt Holland finished with 13 to lead Air Force.