The Lobos, who have already clinched the Mountain West Conference title, won their sixth straight game.
Nevada held a one-point lead with 8 minutes to play and then went 5 minutes without a point.
Bairstow put New Mexico in front to stay with the first of his consecutive layups and he finished with 15 points and 10 rebounds.
New Mexico (26-4, 13-2 MWC) was led by 25 points from Tony Snell, who made 8-of-11 from the field
Malik Story also had 25 points for Nevada (12-17, 3-12), which has lost six straight.