Kirkegaard's TD pass capped a nine play, 75-yard drive after Air Force had taken a 13-10 lead on its previous possession.
Kirkegaard finished the game completing 23 of 37 passes for 282 yards and no interceptions.
Dominic Rufran caught six passes for 119 yards.
Wyoming (2-0, 1-0 Mountain West) earned its 300th home victory in the win.
The Falcons, 1-1, 0-1) were led by quarterback Kale Pearson, who completed 17 of 32 passes for 226 yards and rushed for 22 more.
Jacobi Owens had 76 yards on 18 rushes and one TD for Air Force.
Using a mixture of passes and runs, Air Force drove into Wyoming's side of the field in its first five drives of the game, grabbing a 10-0 lead on a 50-yard field goal by Will Conant and a 3-yard TD run by Jacobi Owens. The Cowboys defense prevented further damage with some timely sacks and stops
The Wyoming offense struggled with poor field position for much of the first half, but finally got a spark when Kirkegaard completed a 41-yard pass to Dominic Rufran. Six plays later, Kirkegaard connected with Jalen Claiborne on an 11-yard TD pass to pull within 10-7 heading into halftime.
After a scoreless third quarter, Wyoming tied the game on a 33-yard field goal by Stuart Williams. The Cowboys scoring drive came after the Falcons' 220-pound senior fullback Broam Hart fumbled on a 4th and 1 play in a violent collision with Wyoming's 246-pound linebacker Jordan Stanton at Air Force's 42-yard line.
Air Force regained a 3-point lead with 3:25 left in the game on an 18-play, 60-yard drive that led to a 32-yard field goal by Conant.