Chris Fouch added 19 points, and Frantz Massenat scored 11 for the Dragons, who are 15-12 overall and 7-7 in the CAA with two games remaining. Guards Devon Saddler and Davon Usher scored 20 and 19 points, respectively, for the Blue Hens.
Delaware, which still owns a one-game lead over Towson with two left, is 20-9, 12-2.
Allen's offense was needed even more since Massenat collected four fouls and sat out 10 minutes of the game.
"When Frantz went out, I knew I had to step up and get everybody involved, so that is what I tried to do," said Allen, who shot 10 for 17 from the field and had a team-high four assists.
A 6-foot-7 sophomore, Allen is now averaging 12.7 points, and against Delaware he came off the bench to provide instant offense.
"Tavon stepped up big-time and made shots when we needed them and carried our team today," Massenat said.
Trailing by 63-62, Drexel scored six unanswered points on two free throws by Massenat, a driving layup by Allen, and two more foul shots by Massenat with 39.1 seconds left.
Delaware's Usher hit 2 of 3 fouls shots with 33.2 seconds left, and Drexel's Rodney Williams made 1 of 2 free throws, increasing the lead to 69-65 with 28.1 seconds remaining.
After a missed shot, a jump ball gave Delaware possession with 18.7 seconds left. Kyle Anderson missed two three-pointers, and Drexel ran out the clock.
Delaware guard Jarvis Threatt, who scored 18 points in the Blue Hens' 90-77 win at Drexel on Jan. 20, and forward Marvin King-Davis, missed their eighth game due to a suspension for a violation of athletics department policy.
Then again, Drexel's Damion Lee and Kazembe Abif are out for the season with injuries.
Delaware coach Monte' Ross wasn't about to make excuses for his missing players.
"You are going to have some game when a team is better than you that day," Ross said. "I think Drexel was better than us today."