Utah State (2-5) beat a Division I-A team for the first time this season. The Aggies' only other win was against I-AA Idaho State.
Freshman Dontrell Moore ran for 164 yards, his first 100-yard game, to lead the Lobos (3-5).
After Utah State got its overtime touchdown, New Mexico took its turn from the 25, and the Lobos found the end zone when Casey Kelley threw a 12-yard pass to Hank Baskett. The game appeared headed for a second extra period when Byrd's kick went wide right.
The Lobos seemed to have the game wrapped up when Terrell Golden returned an interception 59 yards for the score to put his team ahead 28-21 with just 25 seconds left.
New Mexico had tied the game less than a minute earlier, when Kelley completed a 95-yard drive by hitting tight end Bryan Penley with a 9-yard TD pass.
Fuentes completed 16 of 34 passes for 340 yards, while Kelly was 18-of-25 for 202 yards.
An 80-yard touchdown pass from Fuentes to Kevin Curtis in the opening drive of the third quarter gave the Aggies their first lead of the game. In the fourth quarter, Curtis caught a 39-yard pass from Fuentes to put the ball at the New Mexico 10-yard line. Three plays later, a 5-yard run by Richard Watson gave the Aggies the lead again.
Utah State had lost three straight to New Mexico, including a 39-36 triple-overtime defeat when the teams last met, four years ago in Albuquerque.