The Blazers (19-7, 7-2) trailed 31-26 at halftime but opened the second half on a 9-0 run to take the lead for good. Another 9-0 run put UAB up 47-37 with 13:01 remaining and the Owls (7-15, 1-8) got no closer than eight points the rest of the way.
Johnson was 7 of 10 from the field and 11 of 13 from the free-throw line for UAB. He also had five assists and three steals. Jamarr Sanders added 18 points — all in the second half — and Elijah Millsap 13 for the Blazers.
Arsalan Kazemi led Rice with 15 points and 13 rebounds. Lawrence Ghoram added 13 points and Trey Stanton had 12.
UAB had a big advantage at the free-throw line, hitting 28 of 34 shots compared to just 10 of 18 for Rice.