With Air Force trailing 64-63, Lyons hit the go-ahead jumper with eight seconds left following a New Mexico State turnover and then sank 1 of 2 from the line with a second remaining.
The Falcons (4-2) led through most of the second period and were up 57-55 with 1:46 to play in regulation. With two seconds left, New Mexico State's Jalyn Pennie tied it 59-59 on a free throw, sending the game into overtime.
Lyons finished with 17 points, five rebounds and five assists for the Falcons. Hayden Graham scored 16 points with seven rebounds, Zach Kocur had 15 points with six rebounds and Kyle Broekhuis added 11 points.
Pascal Siakam led the Aggies (4-2) with 26 points and 12 rebounds.