The win for the Aggies (2-2) in the teams' 100th meeting breaks a six-game losing streak to UNM (0-4).
Fleming came off the bench for the Aggies to complete 4 of 5 passes, hitting Anderson on a fourth-and-goal situation.
Seth Smith gained 113 yards on 31 carries for the Aggies, offsetting the career-high 98 yards that New Mexico's Dennard Dennis totaled.
The Lobos moved into field-goal position in the final seconds, but James Aho's 47-yarder that would have tied the game was wide right. It was his second miss of the night.
Will Grier threw four touchdown passes to four receivers -- all in the first half -- and No. 25 Florida downed third-ranked Mississippi 38-10 on Saturday night.
Leonard Fournette highlighted his SEC-record third straight 200-yard game with a 75-yard touchdown run, and No. 9 LSU overcame Eastern Michigan 44-22 on Saturday night.
An officiating error led to an Oklahoma State touchdown Saturday in the Cowboys' 36-34 win over Kansas State.
Florida State running back Dalvin Cook exited in the first quarter of the Seminoles' 24-16 win over Wake Forest with a left hamstring injury. He is day-to-day.
As if things weren't going bad enough at Texas, the Longhorns appear to have players on Twitter during lopsided losses and jabs at their coach accidentally being sent from other nearby teams.
Tulane long-snapper Aaron Golub became the first legally blind person to play in a Division I football game Saturday, according to the school.