LOUISVILLE, Ky. — After San Diego State Coach Brian Dutcher snipped the last two strands of the net, he held it aloft like a trophy to admire for his players, coaches, staff and the hundreds of fans who let out a celebratory roar for the Aztecs’ first trip to a men’s Final Four.

Then Dutcher pivoted atop the ladder, placed the scissors in a receptacle, and with his arms outstretched, leaned back until he was free-falling into the arms of his players.

The Aztecs were as sure handed with their coach as they were with the opportunities they’ve been presented along the path to Houston for next weekend’s Final Four.

They took advantage of a referee’s kind whistle on Sunday, with Darrion Trammell sinking a free throw with 1.2 seconds left that was the difference in a 57-56 victory over Creighton. They had also made the most of their chances against top-seeded Alabama on Friday and the good fortune of Furman clearing fourth-seeded Virginia out of their path in the first round.

In that way, San Diego State has much in common with the other participants of this haphazard, helter-skelter Final Four: Miami, Florida Atlantic and Connecticut.

For the first time since 1970, the Final Four will have three first-time participants. And though Connecticut is gunning for its fifth championship since 1999 (under its third different coach), the Huskies began their run with only one tournament victory since their last title in 2014.

“It’s March Madness, emphasis on the madness,” Matt Bradley, a senior guard, said in a jubilant San Diego State locker room.

A sign of what was to come arrived in the First Four in Dayton, Ohio, when Fairleigh Dickinson, a No. 16 seed that didn’t win its conference in the regular season or the postseason tournament, won a play-in game, which served as a prelude to their upset of No. 1-seeded Purdue.

“Winning our first game and seeing Purdue lose, that gives teams hope,” Bradley said. “OK, anybody can be beaten. What teams did during the regular season doesn’t really matter at this point. It’s just about these games now. I think we took that challenge on. Whoever we play, let’s not worry about what they did before. Let’s see who has the best guys now.”

And so, the Aztecs will face the ninth seed in the East Region, Florida Atlantic, which plays its home games in cozy 2,900-seat Baldwin Arena, a generously named gym. The Owls, whose nickname is derived from their Boca Raton, Fla., campus being designated a burrowing owl sanctuary in 1971, will play in front of far more people on Saturday than they did in their 17 home games this season.

Florida Atlantic’s journey to Houston began in the same vein as San Diego State’s clinching moment, with a debated officiating decision. In the first round against Memphis, there was a tied-up ball after the Tigers were denied a timeout, which led to a floater by Nick Boyd with 2.5 seconds left that gave the Owls a 1-point victory. They then rallied late to beat Fairleigh Dickinson, Tennessee and Kansas State.

Miami’s hopes were also almost over as soon as they began. The Hurricanes trailed Drake, 56-53, with two and a half minutes remaining before they scored the game’s final 10 points. Since then, they dominated No. 4-seeded Indiana, thumped No. 1-seeded Houston and rallied from a 13-point, second-half deficit on Sunday to oust No. 2-seeded Texas.

The combustible Canes, who had to sort out sore feelings over the summer when Nijel Pack transferred in from Kansas State on a $800,000 endorsement agreement from a booster — prompting his new teammates to want to renegotiate their deals — endured a sideline spat on Sunday between guard Harlond Beverly and Coach Jim Larrañaga.

It put to the test the Coach Feel Good beliefs of Larrañaga, 73, who is an avid reader of Eastern philosophy and motivational quotes and subscriber to cleansing breaths.

After Sunday’s comeback, he cited Stephen Covey’s “Seven Habits of Highly Effective People,” noting that habit No. 2 is to begin with the end in mind. “The first day of practice, we’ve got to start visualizing right now what we want to accomplish and be working toward that every single day,” he said.

One other trait will set Larrañaga apart this week: He’ll be the only one on the sidelines in Houston who has previously been to a Final Four as a head coach. In 2006, he took 11th-seeded George Mason on a bracket-busting run included the Patriots knocking off top-seeded Connecticut in overtime in the round of 8.

Miami will have a chore on its hands on Saturday night when he sees Connecticut again.

The Huskies have breezed through the tournament, dispatching Iona and St. Mary’s with second-half blitzes and then blowing out Arkansas and Gonzaga. Their average margin of victory is 22.5 points.

The only difficulty Connecticut has endured has come off the court. When they arrived in Las Vegas, some of the players’ rooms at the Luxor hadn’t been cleaned after the previous guests had left, prompting the team to move to Resorts World. “It was a disaster,” Coach Dan Hurley told the Hartford Courant.

Then, when the Huskies practiced at the University of Nevada-Las Vegas, their team buses were burgled. Donovan Clingan, a freshman center, had his iPad stolen and a laptop had also gone missing.

Hurley, whose father Bob Hurley Sr., was a prominent high school coach in Jersey City, N.J., and whose brother, Bobby, was a star at Duke, was hired to restore the luster to Connecticut after Kevin Ollie — who coached the Huskies to a national title in 2014 — was fired. He had just led Rhode Island to the second round in back-to-back seasons.

It took five years for the Huskies to win a tournament game.

“It becomes a little bit of a mental hurdle,” Hurley said late Saturday night in Las Vegas, referring to getting knocked out in the first round the previous two seasons. “Especially like early rounds of N.C.A.A. tournament where you feel like maybe the burden of the history.”

A little belief, then, can carry a team a long way — all the way to Houston, in fact.

That was clear at the other regional sites, too; at Madison Square Garden; Kansas City, Mo.; and Louisville, Ky.

Just before Dutcher climbed the ladder, the San Diego State fans put a twist on a familiar chant that echoes through their rowdy home arena, one they borrowed from fans of the United States national soccer teams. All they did was alter their message of hope from the future simple tense to the past.

“I believe that we have won,” they roared over and over.

A short while later, Dutcher put his own faith to the test, his players rewarding his trust one more time this season.