SAN ANTONIO — Villanova, leading by 6 points in a game that felt closer with eight minutes remaining, was patient and precise. Players zipped the ball around and across Michigan defenders, waiting for the ideal scoring opportunity. Finally, Caleb Daniels, Villanova’s redshirt senior guard, grabbed the ball in the paint and drove forward, his body running into the 6-foot-11 Wolverines forward Moussa Diabate as the ball went through the hoop. The sound of an official’s whistle followed, setting up a 3-point play.

The Wildcats didn’t join their fans in celebrating, though. They got back on defense. The No. 11-seeded Wolverines, like they had all evening, were sure to respond. And with a pair of jump shots and some free throws, Villanova’s lead, once 9 in the half, was 4.

But No. 2-seeded Villanova, after playing far from its best game, got a driving layup by Jermaine Samuels and a 3-pointer on its next possession, establishing enough of a cushion to get past the Wolverines, 63-55, on Thursday in the round of 16 in San Antonio.

Coach Jay Wright’s Wildcats, one of two Big East teams left in the N.C.A.A. tournament along with Providence, have shown that the conference is all but theirs until another program pries it out of their 3-point-shooting hands. But any team that was ranked as high as No. 4 in the regular season would be looking for more than a conference championship, which the Wildcats won in Madison Square Garden this month for the fourth time since 2017.

“We’re beat up. We’ve got to rest up,” Wright said. “We can learn a little bit watching film. It’s about surviving now.”

Villanova took down Michigan in the national title game in 2018, but has had worse luck in the tournament since: In 2019, the Wildcats were defeated by Purdue in the second round, and last year they were knocked out in the round of 16 by Baylor, the eventual champion.

But now, the Wildcats are one win away from their seventh Final Four appearance in program history during a run in which they have won two national championships since 2016 and produced formidable N.B.A. talent.

Donte DiVincenzo of the Sacramento Kings, Mikal Bridges of the Phoenix Suns, Jalen Brunson of the Dallas Mavericks and more have come through the program in recent years.

Collin Gillespie, Villanova’s sharpshooting graduate student guard, will enter the upcoming draft after leading the Wildcats with nearly 16 points per game this season and stationing himself as one of the best 3-point shooters in men’s college basketball. Gillespie hit four of 10 3-point attempts against Michigan, playing all but one minute and finishing with 12 points.

Villanova, as a team, didn’t shoot the ball well from beyond the arc. Of 30 3-point attempts, Villanova made nine, while the Wolverines were 6 of 18 from deep.

Michigan surprised many by reaching this round of the tournament, given the Wolverines’ up-and-down season, which included Coach Juwan Howard’s five-game suspension after a postgame skirmish with a Wisconsin coach. But the Wolverines had the size advantage they needed to be successful against the smaller Wildcats. Michigan went down low to its leading scorer and big man, Hunter Dickinson, early, and he found success there, finishing with 15 points and 15 rebounds despite being in foul trouble. Villanova inserted a smaller lineup and drew two quick fouls on Dickinson in the first half, and he finished with four fouls.

“We didn’t run into anybody anywhere like Hunter Dickinson,” Wright said after the game. “Man, this dude is a handful. He’s got size, girth, strength, skill, intelligence, competitiveness. We run into some good ones in the middle, but I think he’s the best we’ve run into this year.”

But on a mediocre shooting night for both teams, Michigan missed too many of the shots that mattered, including, uncharacteristically, the ones around the rim and half of its free throws.

“We got good looks,” said Eli Brooks, Michigan’s graduate student guard. “We just didn’t capitalize. I think we got the looks that we wanted. We just didn’t make the shots.”

Coach K is still in the hunt for a sixth championship.

Duke (30-6) survived and advanced against seventh-seeded Michigan State on Sunday, extending Coach Mike Krzyzewski’s career for at least another week. Still, the Blue Devils will have their hands full with an older Texas Tech team ranked No. 1 in defensive efficiency by KenPom. The Red Raiders (27-9) are allowing just 13 points in the paint per game in the tournament, the fewest in the field. They will look to play a physical game that disrupts Duke’s offensive flow (Thursday, 9:39 p.m. Eastern, CBS). The Blue Devils feature five potential N.B.A. draft picks, led by the freshman forward Paolo Banchero, who averages 17.0 points and 7.9 rebounds per game.

“I’m not that excited,” Mark Adams, Texas Tech’s first-year coach, said of facing Duke. “I don’t know if anybody is excited about playing Duke.”

Krzyzewski has “always been a mentor of mine, someone I looked up to,” Adams added. “Not only is he a great coach, but a great person. Just done so much for basketball. And he’s built a program which we all admire and respect. It’s just one team to the next, he’s got a dynasty he’s built. So our hat’s off to him.”

Should Duke advance, it could get a rematch against Gonzaga in what would be the highest-profile game of the tournament. The Blue Devils beat the top-ranked Bulldogs, 84-81, on Nov. 26 in Las Vegas with Banchero scoring 21 points and the junior forward Wendell Moore having 20 points, 6 rebounds and 6 assists.