Bairstow's two inside scores coming down the stretch erased a 48-45 deficit and Tony Snell sealed the contest for the Lobos (21-4, 8-2 MWC) on a jumper with 20 seconds to play.
Kendall Williams provided 13 points while Snell and Alex Kirk chipped in with 10 each as New Mexico bounced back from a Saturday loss to Nevada-Las Vegas.
Fresno State (8-15, 2-8) was led by led by Kevin Foster's double-double of 10 points and 10 rebounds.
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