Freshman Nick Sherry completed 23 of 32 passes for 250 yards and a touchdown, and the Rebels (1-3, 1-0 Mountain West) held off the Falcons in the final seconds at midfield.
The Rebels trailed 28-17 at halftime before scoring 21 unanswered points on Randle scoring runs of 1 and 9 yards and William Vea's 7-yard TD reception with 6:56 remaining.
Cody Getz rushed for 113 yards and a touchdown for Air Force (1-2, 0-1), and quarterback Connor Dietz ran for 68 yards. He was 3 of 9 for 90 yards passing.
The Falcons finished with 352 yards rushing.
Air Force cut the UNLV lead to 38-35 on a Dietz 3-yard run with 4:41 remaining. It got the ball back with 51 seconds left and no timeouts at its 25 and advanced 20 yards before running out of time.
Air Force played before 112,522 spectators at Michigan in its previous game two weeks ago, a much bigger gathering than the 14,054 in attendance on Saturday at Sam Boyd Stadium.
UNLV lost three home games by a combined 14 points against Minnesota, Northern Arizona and Washington State.
In the series, Air Force had won five straight and nine of 10.
The Falcons out-rushed UNLV 180-7 in the first quarter and led 14-7. Mike DeWitt had a 1-yard run, and Getz added a 2-yard run with 25 seconds left in the quarter. The Rebels' score was set up by a 60-yard pass from Sherry to Marcus Sullivan to the Air Force 2-yard line, and Cornett ran it in on the next play.
After the Rebels cut the lead to 14-10, Alex Means' 21-yard interception return gave Air Force the ball at the UNLV 8. Two plays later, Broam Hart ran it in with 7:10 remaining in the half. UNLV cut the lead to 21-17 on a Randle 1-yard run with 50 seconds left.
Air Force scored on a Dietz 7-yard run through the left side with 10 seconds remaining in the half. Dietz's 58-yard pass to Ty MacArthur set up the score.