Harris scored on a putback to give UTEP (16-7, 8-3 Conference USA) a 64-63 lead. C.J. Cooper extended it to three on two free throws with 18.8 seconds left.
FIU called timeout with 2.4 seconds left and Mavin missed an off-balance 3-point attempt from the top of the key as time expired.
UTEP led 57-47 with 8:10 left, but the Golden Panthers rallied to take their first lead at 63-62 on Mavin's driving basket with 1:23 left.
Mavin had 23 points and Daviyon Draper added 21 to lead FIU.