Miami, trailing and listless for much of the first 30 minutes of its round-of-8 game against Texas on Sunday, mounted a stunning, compact comeback on Sunday in the N.C.A.A. tournament to claim the last spot in the Final Four next weekend in Houston.

Miami, which won 88-81, will play in its first national semifinal, with San Diego State, Florida Atlantic and Connecticut, its next opponent, rounding out the rest of the field.

The Hurricanes got off to a fast start but quickly fell behind the Longhorns, who were playing before an overwhelmingly pro-Texas crowd.

Miami, calm and persistent, began creeping back from a 13-point hole with less than 14 minutes left, setting up a furious few minutes when it finally strung together a series of defensive stops and a run of key free throws. Norchad Omier, a sophomore forward, hit two of them when the game was tied with a minute left.

Miami was plodding, eventually mounting that 9-point run when it seemed like the Longhorns would cruise to a home-state advantage in Houston next weekend.

Both teams featured players that typify the mobility and commercialization of modern college basketball. Texas started four transfers, including its hard-charging graduate point guard Marcus Carr.

Miami was led by Jordan Miller, a 6-foot-7 senior who scored 27 points and had a crucial steal with two minutes left. The team was also paced by two guards, Isaiah Wong and Nijel Pack, who played in Coral Gables, Fla., this season in part because of lavish name, image and likeness arrangements worth hundreds of thousands of dollars. When Miami had the ball on Sunday, it was often held by Pack, who scored 15 points by drifting in and out of scoring lanes, soaring to the hoop for delicate finishes and stepping back for short fadeaways.

San Diego State slipped past a familiar opponent.

Back in November, when Creighton set out for the Maui Invitational, the Bluejays stopped in San Diego and the next day shared a charter jet to Hawaii with the San Diego State team. Memories of Creighton’s overtime win over San Diego State in the first round of the men’s N.C.A.A. tournament the previous March might have led to some awkward moments.

But the two coaches, Greg McDermott of Creighton and Brian Dutcher of San Diego State, sat across the aisle from each other, poring over film on their laptops, trading scouting reports and ruminating about the possibility of playing each other again in the early-season tournament.

They did not, at least then. And when the teams flew back to San Diego, dropping the Aztecs off, the coaches — and their teams — bid each other adieu.

See you down the road.

That meandering road carried both teams to a place they had never been before, a regional championship in which the coaches and players — including two brothers, Creighton’s Arthur Kaluma and San Diego State’s Adam Seiko — marveled at the serendipity of it all.

“I never thought we’d be playing them here or I would have tried to steal a few play calls off his computer,” Dutcher mused.

The next time the two coaches get together, the bounds of fraternity will be tested after San Diego State rallied for a 57-56 victory on Darrion Trammell’s free throw with 1.2 seconds left. The game was so thick with turns and tension that it did not end until well after what turned out to be the final buzzer.

San Diego State, which had only twice before been to the second weekend of the N.C.A.A tournament, will play ninth-seeded Florida Atlantic, the champion of the East Region, on Saturday in Houston with a berth in the national championship game on the line.

The deciding play came as San Diego State ran the clock down for the final shot.

Trammell drove to the lane with Ryan Nembhard on his right hip and let loose a floater that just grazed the rim as the buzzer sounded. But rising above the cacophony, as Trammell lay on the court, was the whistle of the referee Lee Cassell.

Soon, Trammell went to the foul line with the crowd on its feet, four teammates behind him — and the entire Aztecs bench — locking arms. Trammell’s first free throw rolled off the rim, and the crowd’s roar grew even louder.

He took two dribbles, a deep breath and swished the next one.

Baylor Scheierman, who played quarterback in high school, inbounded the ball and threw a long pass toward Kaluma and San Diego State’s Aguek Arop near the other endline. They tipped the ball out of bounds as the buzzer sounded. The officials, though, waved both teams back to their benches and reviewed the play to see who had touched the ball last and whether there was any time left.

After a few minutes, they ruled that time had expired. San Diego State players sprinted onto the court to celebrate.

The final six seconds “felt like an eternity,” Arop, the San Diego State forward who is from Omaha, said on the court after he and his teammates cut down the nets.

McDermott, who shouted at the officials as they left the court, said he did not get an explanation of the ruling that time had expired. The N.C.A.A. said in a statement that the review indicated that the clock started late. McDermott declined to criticize the foul call.

“Two teams played their tails off,” he said. “Officiating is part of the game. We’re not going to go there. We lost a game because we didn’t do enough and San Diego State did.”

Dutcher, noting that he was an assistant at Michigan when it beat Seton Hall for the national championship in 1989 on a controversial foul call, appreciated McDermott holding his tongue. “It’s hard. That’s what we all do is have some grace in losing even though we may not agree with the call,” Dutcher said.

San Diego State guard Lamont Butler had 18 points on 8 of 11 shooting, and was set to take the final shot until Creighton fouled him with six seconds left, forcing the Aztecs to take the ball out of bounds but also turning off the shot clock.

Trammell, a transfer from Seattle University who had scored 21 points to help carry San Diego State to an upset of top-seeded Alabama on Friday, had made only 5 of his 14 shot attempts and had not been to the free-throw line until the final second.

When he stepped to the line after missing the first attempt, Trammell said he reminded himself that he’d shot 1,000 free throws in the last week and that the moment was not too big for him. “I just had to believe in that,” he said. “Just having that confidence that, yeah, I missed the first one, but I definitely wasn’t going to miss the second one.”

San Diego State, which enjoys a raucous home-court advantage, has been an N.C.A.A. tournament regular out of the Mountain West Conference, but it has been cast in the shadow of Gonzaga and the Pac-12 Conference teams of the moment. Nevertheless, this is a moment that the program has long believed would come.

“You picture the belief when you sleep, you picture the belief when you work out, and you hope that the dream comes true,” said Nathan Mensah, San Diego State’s 6-foot-10 senior center, who contributed 8 points, 6 rebounds and 3 blocks. “Finally that dream came true for us.”

The game unfolded as a contrast of styles, Creighton’s artful, free-flowing offense finding its way against San Diego State’s muscular, methodical brand of basketball.

It was played largely at San Diego State’s preferred pace, but Creighton played almost exclusively with the lead, fending off repeated Aztec charges until the final minutes.

After Creighton surged to a 28-20 lead, urged on by a largely blue-clad crowd, San Diego State finally figured out how to keep the Bluejays’ 7-foot-1 center, Ryan Kalkbrenner, from having his way diving to the rim for alley-oops or putting his nifty post moves to use. Mensah did yeoman’s work against Kalkbrenner, who led Creighton with 17 points.

When Trammell sank a jumper near the free-throw line, San Diego State had finally gotten even with less than three minutes left in the half. But Creighton didn’t allow another basket and carried a 33-28 lead into the break.

San Diego State came storming out of the locker room, and when Mensah blocked consecutive shot attempts, sparking a fast-break lay-in by Butler, the Aztecs had their first lead, 35-34.

Again, it did not last long. Kaluma answered with a driving layup and San Diego State went cold, missing its next 10 shots — many of them on drives to the rim. But the Aztecs leaned on their defense and depth, wearing down the Bluejays after halftime. Creighton shot just 28 percent in a second half in which it tied a season-low 23 points for a half. The Bluejays missed all 10 of their 3-point attempts in the half.

The parents of Kaluma and Seiko, along with their two young sisters, sat a few rows up at center court wearing customized white T-shirts with a basketball, both schools’ logos and the names and numbers of the brothers.

When the game was over, their two sons exchanged a hug in the handshake line and Seiko told Kaluma he loved him. In that moment, they also acted as exemplars for their teams, one of which cut down the nets while the other felt like it had its hearts cut out.