Leftwich completed 20 of 39 passes for 271 yards - also a career best - in his first action since suffering a concussion against New Mexico State on Sept. 19. Batson finished with three receptions for a career-high 143 yards.
FAU (1-6, 1-3 Conference USA), which came in averaging 23.2 points per game, scored all of its points in the second quarter to take a three-point lead into the half.
Leftwich and Batson connected for a 30-yard touchdown midway through the third quarter to put UTEP (3-4, 1-2) in front 21-17. Jay Mattox added field goals of 35 and 42 yards in the fourth for the Miners.
Trey Rodriguez finished with 114 yards rushing and a score for the Owls.