Chris Fouch added 16 points for the Dragons (29-6), who host Massachusetts in a quarterfinal game on Tuesday.
"It feels good," Givens said. "[Coach Bruiser Flint] talks about I'm the senior and it could be my last game. You always want to keep playing. I'm happy."
Johnny Moran scored 19 points to lead Northern Iowa (20-14).
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Drexel, one of the last teams left out of the NCAA tournament field, led by as many as 18 points in the first half but Moran's 3-pointer with 34.6 seconds left closed the margin to 61-60. Givens was fouled out of a timeout and made 1 of 2 free throws with 24.3 seconds to go, giving the Dragons a 62-60 lead.
Anthony James missed a driving layup at the other end before Damion Lee hit two free throws with 12.9 seconds left to make it 64-60.
Chip Rank gave the Panthers life on a 3-pointer with 6.1 seconds left before Frantz Massenat was fouled and made 1 of 2 free throws.
Moran's last-second 3-point attempt went off the back of the rim.
"It looked good," Northern Iowa coach Ben Jacobson said. "It had great rotation on it. He had to double-pump but he still got a good shot off. It looked like it was right on. I thought it was going to go in."
Flint said the Dragons got away from their game plan in the second half. Drexel wanted to limit the Panthers to shots inside the arc since they came in shooting 38.5 percent from 3-point range.
"I kept telling these guys the reason why these guys are in the second round of the NIT is they're a pretty good team," Flint said. "We wanted them to shoot layups. We totally lost our concentration in the second half of what we wanted them to do."
The Panthers shot 10 for 23 (43.5 percent) from beyond the arc, including 7 for 14 after halftime against a Dragons team that led the Colonial Athletic Association and ranked 10th in the country in 3-point defense (37.6 percent) this season.
Massenat also finished in double-figures for Drexel with 11 points.
Drexel won despite being outrebounded 32-24, just the seventh time this season the Dragons have lost the battle of the boards. The Dragons entered 22nd in the country in rebounding margin, bettering opponents by 6.0 per contest.