The Irish led by as many as 13 late in the first half and by nine, 56-47 with 12:44 remaining but the Blue Demons embarked on an 11-1 run over the next five minutes, eventually taking a 63-60 lead, their first advantage since the 13:51 mark of the first half.
The Irish took back the lead late thanks to consecutive determined defensive stands and penetration by Grant, who first hit for a bucket and foul on a drive to the bucket, then found backcourt mate Eric Atkins for a three-point shot and 66-63 edge.
DePaul though wouldn't fold, tying the contest at 66 thanks to a Cleveland Melvin jumper and subsequent split at the free throw line by Brandon Young. Young's was the final point of regulation and resulted in the second straight game between the teams to need an extra session.
The Irish famously played five overtimes in their last outing, a 104-101 win vs. Louisville Saturday.
"I'm thrilled we escaped that one. But really give DePaul credit," said Irish head coach Mike Brey. "They're hard to guard for us. They were in Chicago (a 79-72 overtime win), and we had to escape. And they were in the second half tonight also. Their ball pressure defensively really rattled us, and we couldn't get much in the second half."
Notre Dame committed an uncharacteristic 15 turnovers, a number that helped offset their decisive 44-25 advantage on the glass.
DePaul drilled eight of 16 three-point field goals and committed just seven turnovers of their own. The Irish struggled to contain the mid-range maven Melvin throughout. The junior forward hit for 23 points on 11-21 shooting but missed the potential go-ahead jump shot with the score tied, and 24 seconds remaining in regulation.
Melvin's classmate Young added 20 while Donavan Kirk hit for 16, also on a variety of mid-range shots.
"I think we turned the ball over a little bit. They got some easy buckets," said junior point guard Eric Atkins who scored 13 with 9 assists. "They just started hitting their shots. They just really got going offensively. They are a very good offensive team, and they proved that tonight. Down the stretch we finally got a few stops to get the win."
Those defensive stops were largely due to improved effort by the Irish big men defending the pick-and-roll up top.
"It's definitely been the focus (in practice), a lot on ball screens because we've had a few breakdowns where we weren't guarding screen well and people were splitting us," said senior forward Tom Knight. "Every day before the game we decide if we're flat, showing, or hard hedging and we just in practice on that. I feel like everyone has been doing much better as of late."
Notre Dame showcased its new 8-player rotation -- half of which are players 6'10" or taller. Backup bigs Garrick Sherman and Zach Auguste both played well off the bench, combining for 13 points and seven rebounds on 6 of 8 shooting over 24 total minutes.
Despite a 20-7 edge in bench scoring; despite a 14-6 advantage in second-chance points, and despite an alarming 21 more free throw attempts (Notre Dame was 21-28, DePaul 4-7), the Irish still needed five extra minutes to secure victory.
They're 3-1 this year in extra sessions (losing to Saint Joseph's in November) and were 3-0 last season. Its a source of pride among the regulars.
"We have learned that we are in great shape and that we can play through being tired," said Cooley. "We know all of our situations because we go through them a hundred times in practice. We just have to go out there and play well."
Playing well for 40 minutes rather than in spurts for 40-plus is the next step.