The Red Raiders (2-0) benefited from two blocked punts at the beginning of each half that led to short touchdown drives.
The Lobos (0-2) didn't help themselves, committing 16 penalties that cost them 145 yards.
Potts, who completed passes to nine different receivers, went 22-34 in the air, with Leong catching five for 75 yards.
Baron Batch also rushed for two touchdowns for Texas Tech.
The Red Raiders took a 24-3 lead early in the second quarter before the Lobos closed within 31-17 and had a chance for more points before halftime, but an offensive pass interference penalty and a personal foul on the same play negated a UNM touchdown with seven seconds left.
Texas Tech scored 45 seconds into the game on a 55-yard pass from Potts to Austin Zouzalik.
Aaron Crawford then blocked a punt on the Lobos' next possession with Jonathan Brydon recovering at the 3. Eric Stephens scored on the next play for a 14-0 Red Raiders' lead five minutes into the game.
After a New Mexico field goal on the next possession, Stephens returned the kickoff 93 yards to the UNM 7, with Batch scoring two plays later.