Denzell Watts had 12 points for UAB (14-13, 10-4 Conference USA). Hakeem Baxter scored 10 and Chris Cokley added 10 points and 10 boards.
UAB trailed 38-35 five minutes into the second half, but runs of 10 and nine points stretched the lead to 16 with 5:31 to go.
Western Kentucky closed to six with 2:35 after an eight-point spurt, and Aleksej Rostov's 3-pointer made it a three-point game with 35 seconds left.
Norton made a pair from the line to push it back to five, and on the next possession WKU's Chris Harrison-Docks missed a 3-pointer. Cokley and Watts both made free throws to seal it.
The Hilltoppers led by 10 in the first half, but UAB closed on a 9-2 spurt to trail 31-28 at the break.
T.J. Price led Western Kentucky (16-9, 9-4) with 22 points.