New Mexico handed the Bearcats their first loss of the year, getting a blocked shot by Alex Kirk on a 3-point by Sean Kilpatrick with 4 seconds left to preserve the victory.
After Snell's jump shot gave the Lobos the lead, Cincinnati missed its final three attempts from the field.
The game was close throughout with Cincinnati holding a four-point lead at halftime and New Mexico taking as much as a six-point advantage in the second period.
Cashmere Wright gave Cincinnati its last lead at 54-53 when he made a 3-pointer with 2:54 to go.
After a time out, Snell hit his jumper that turned out to be the final points of the game.
New Mexico, which improved to 13-1, was led by Kendall Williams with 15 points.
Cincinnati had won its first 12 games of the season and its loss leaves Duke, Michigan, Wyoming and Arizona as the only unbeaten teams in major college basketball.
Kilpatrick and JaQuon Patrick both scored 15 points for the Bearcats, who hit just 31 percent of their shots from the field. Cincinnati had a 13-game home winning streak snapped.