Dozier had 15 points and both Shawn Taggert and Tyreke Evans had 12 for the Tigers (6-2), who were coming off an overtime loss last Saturday at Georgetown.
Memphis made only a third of its field goal attempts, but the Tigers still put together a 48-30 lead with 9:30 to play in the game.
UALR then went on a 15-3 run that brought the Trojans within six with 2:30 left, but they could manage just one more field goal the rest of the way. Memphis hit eight free throws in 12 attempts down the stretch to maintain its lead.
The Trojans (7-3) were led by Steven Moore with 16 points. They have lost three out of five after opening the season with five straight wins.