Nick Norton scored 14 points, Chris Cokley and Robert Brown added 13 apiece and Toshin Mehinti grabbed 10 rebounds for the Blazers (18-4, 8-1).
Trailing 40-32 at halftime, Marshall gained on a layup and dunk by Ryan Taylor to pull within 49-43, then trailed 56-54 after Taylor's jumper with 11:35 left.
UAB led 79-72 on Cokley's jumper with 2:56 left, but Marshall closed to within 79-78 on Taylor's 3-pointer with 47 seconds to go.
Brown missed a jumper with 15 seconds left, the referees reversed a call that would have given Marshall a final possession after the rebound, and Lee made two free throws to seal it.
James Kelly led Marshall (11-11, 7-2) with 28 points and 10 rebounds and Taylor added 21.