Quigley scored her team's final eight points in the last 2:06 to clinch the win.
Caprice Smith added 14 points for the 10th-seeded Blue Demons (19-11), who advanced to face Rutgers on Sunday night. But Smith fouled out with 4:48 to go and Quigley took over.
"I felt confident knowing my teammates were going to be there to get the rebound and move the ball around," Quigley said.
Charel Allen had 18 points for No. 7 Notre Dame (19-11), which had never before lost in the tournament's opening round. Breona Gray added 17 points in the losing effort.
Notre Dame was up 44-38 when DePaul went on a 19-2 run to take an 11-point lead, 57-46.
Allen broke the Irish scoring drought just before the 10-minute mark. Less than seven minutes later, Allen hit a free throw to complete a 10-0 Notre Dame run and tie the game at 66-66.
"We came all the way back and then missed a bunch of free throws," said Notre Dame coach Muffet McGraw. "We missed three free throws with a chance to take the lead, and you don't expect that when you have your good free throw shooters on the line."
There were five ties and four lead changes in the first half, and neither team led by more than six points. The teams combined for 38 turnovers, 27 in the first half.
A basket by Irish forward Crystal Erwin with 30 seconds left in the opening half gave her team a 36-34 advantage heading into the break.
Notre Dame has made the NCAA tournament for 11 consecutive years, a streak now in jeopardy.
Postgame Notes: With four three-pointers, the Blue Demons set a new single-season record with 236 three's... Allie Quigley's 12-for-12 effort from the line marked the just the third occurrence in school history with a player perfect from the line with at least 12 attempts... she also notched her 75th career game in double-figures and moved into 12th all-time in scoring... the win was No. 590 in program history... DePaul is 2-1 all-time in BIG EAST Championship play... at least four players have reached double-figures in 15 games this season... DePaul's .957 (22-23) free throw percentage is the fifth-best single-game effort in school history... the Blue Demons lead the series, 17-11, and have won two straight over the Irish.
Official Box Score
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