The Fighting Irish (30-5) will play the winner of Texas-Butler on Saturday.
The third-seeded Irish returned to the tournament after a one-year absence determined to build off their ACC Tournament title and make a deep run in March.
The Irish have traditionally struggled in the tournament under coach Mike Brey. They made only one Sweet 16 since 2001.
Northeastern (23-12) gave the Irish fits and threatened to keep Notre Dame's March malaise going. Scott Eatherton had 18 points and David Walker had 15.
The Colonial Athletic Association champion Huskies played with the 40 minutes of grit needed to avoid a crushing defeat like so many other double-digit seeds throughout tournament history.
But the Huskies finally collapsed with sloppy play over the final minutes that ended the upset bid.
The Huskies chased down the loose ball after an inbound pass was wildly thrown away by the Irish. With a chance to tie it a 67-all with less than 10 seconds left, Quincy Ford lost control of the ball — and the Huskies lost their chance at pulling off the shocker.
Auguste, who made 10 of 14 shots, sealed it with two free throws.
All those fans that love rooting for the double-digit seeds to play spoiler in March had to be rooting for Northeastern. But the Huskies were sloppy late, finishing with 16 turnovers.
Eatherton was in a perfect spot on the block and put back a missed three-point attempt to make it 67-65 with 32.5 seconds left. Even with foul trouble, he came up big down the stretch for the Huskies. Eatherton pushed forward with four fouls and his layup at the 2-minute mark made it 65-61, inching the Huskies closer to their first NCAA Tournament win in more than 30 years.
Grant, though, picked Ford clean and went all the way for a dunk that made it 67-61.
Northeastern was fired up from tip as it chased a small slice of March history. Walker pounded his chest in exultation after his three-pointer on the second possession and the Huskies never let Notre Dame's lead grow to more than five points.
Walker's second three-pointer of the half evened the score at 18-all and the Huskies positioned for a 14 seed to win a game for the third straight season.
Caleb Donnelly missed two straight treys that would have given the Huskies the lead. The Irish finally snapped out of their first-half funk and ended the half on a 9-3 run to head into halftime with a 31-27 lead.