Chris Cokley and C.J. Washington added 10 points apiece for the Blazers (10-10, 6-1 Conference USA), who finished shooting 26 of 37 (70.3 percent) from the field and 24 of 26 (92.3 percent) from the line.
UAB built a 10-point lead late in the first period with the help of 3-pointers by Norton and William Lee and went on to a 40-32 advantage at the break.
In the second period, the Blazers' balanced shooting continued and a Norton 3-pointer with 6:22 to play put them up by 11, 61-50. They held on for the win with the help of Norton and Washington, who combined to sink 10 of 10 from the line in the final minutes.
Aaron Bacote led the Monarchs (15-4, 4-3) with 23 points.