The Skyhawks (2-5), who led 50-39 with 4:51 remaining, saw their lead cut to three, 56-53, with 34 seconds left when Cameron Moore made two free throws for the Blazers (1-4).
UAB fouled Liabo with 21 seconds left and he made both for a 58-53 lead. Liabo finished 6 for 6 at the line.
The Blazers had a chance to pull within two points when Jordan Swing got fouled while shooting a 3-pointer with 14 seconds left, but he made just one of his three chances at the line.
Myles Taylor added 13 points and 10 rebounds for the Skyhawks.
Moore and Swing scored 15 points each for UAB. Moore also had nine rebounds and five blocked shots, giving him 26 blocks in five games this season.