Daxter Miles scored 18 points, and Esa Ahmad had nine rebounds.
Carter led the way in matching a season high, while going 8 of 15, including 4 of 5 in 3-pointers.
His last 3-pointer all but finished the fifth-seeded Fighting Irish (26-10) as they attempted one final comeback bid.
Notre Dame's Matt Ryan hit a 3-point basket in the right corner to cut West Virginia's lead to 72-66 with 3:06 left.
Carter responded by crossing through the middle and pulling up to hit a fall-away 3-pointer about 25 seconds later.
"Just staying confident," said Carter, the Big 12's defensive player of the year. "When we step on the court, we feel like we can beat them. We got hot early and kept going from there. We keep that chip on our shoulder."
The Mountaineers continue to overcome the sting of last year's first-round tournament collapse, when they lost to 14th-seeded Stephen F. Austin.
Now they're off to the round of 16 for the first time since 2015, for the fourth time since coach Bob Huggins took over in 2007, and seventh time since the NCAA field expanded to 64 teams in 1985.
Notre Dame was denied a chance to reach the Elite Eight for a third consecutive year .
Bonzie Colson led the Fighting Irish in hitting 10 of 15 shots, scored 27 points and had eight rebounds. Otherwise, the rest of the teammates were stymied.
Guard Matt Farrell was limited to eight points, while V.J. Beachem hit 2 of 14 attempts and finished with 9 points.
Coach Mike Brey said the difference was Notre Dame falling behind 10-0 and eventually getting worn down from attempting to dig out of such a hole. The Irish did eventually cut the lead to 32-29 only to give up two baskets over a 20-second span in a game the Mountaineers never trailed.
"Any time we thought we'd get this thing to four or get it to two possessions, somebody hit a big 3 or they got a putback," Brey said. "It's really spirit-breaking after a while."
Brey had to gamble in keeping Colson in after the star forward picked up his fourth foul with 9:47 left and with West Virginia up 59-47 Colson scored 10 of Notre Dame's next 14 points over a 5-minute span.
"It's horrible," said Colson, whose shoulders were heaving in emotion as he left the court. "It's frustrating when you try to play your tail off and play with everything you have and just leave everything out there."