Wells went 7 for 10 from the floor and led the Terrapins with 10 rebounds. The junior forward also eclipsed the 1,000-point mark for his career.
After missing 12 games with a broken left foot, Allen came off the bench to provide the Terrapins (8-5) with a much-needed spark. Returning from a seven-day layoff, Maryland fell behind 25-10 with 11 minutes elapsed before Allen fueled an 18-4 run.
Still, the Terrapins didn't take the lead for good until Jake Layman drilled a 3-pointer to make it 48-45 with 12:32 left. Nick Faust followed with a jumper from beyond the arc, and Maryland moved ahead 52-47 before Tulsa coach Danny Manning was ejected after receiving successive technical fouls with 10:35 remaining.
Layman made all four foul shots, and Allen added two free throws to give the Terrapins an 11-point cushion. It was the first time in Manning's two years as Tulsa's coach that he was ejected from a game.
James Woodard scored 25 for the Golden Hurricane (4-9). The 6-foot-3 sophomore has led Tulsa in points in seven of the last nine games.
Maryland had five players score in double figures. Evan Smotrycz and Faust each had 13 and Layman added 12 in coach Mark Turgeon's 300th career victory.
Wells and Allen each had nine points in a first half that ended tied at 35.
Tulsa led 11-4 before Wells interrupted the surge with a follow-shot, giving him 1,000 points during a career that began at Xavier in 2011.
Soon after that, Allen entered for the first time this season. The 6-foot-1 sophomore was poised to make his first basket, a breakaway layup, when Woodard swooped from behind and swatted the shot away.
That typified the first half for the Terrapins. It was 13-9 before backup guard Pat Swilling made a pair of 3-pointers during a 12-1 spurt that put the Golden Hurricane up by 15.
And then Maryland finally woke up. Smotrycz sank two jumpers from beyond the arc and Allen scored five points in a run that closed the deficit to 29-28.