The home loss ended a three-game winning streak for the Aztecs and left New Mexico as the only team in the Mountain West Conference without a league setback.
UNLV led by 10 points early in the second half, but San Diego State came back to tie it on a free throw from Jamaal Franklin with 3:43 remaining.
The Rebels outscored the Aztecs the rest of the way 9-2. Birch tallied on a tip-in with 2:22 to play that gave UNLV a 76-73 lead and his layup with 25 seconds to go boosted the Rebels in front 80-75.
Anthony Marshall led the Rebels (15-3, 2-1 MWC) with 20 points, Bryce Dejean-Jones had 14 and Birch added 12.
San Diego State (14-3, 2-1) was paced by Franklin's 27 points.
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