INDIANAPOLIS — Once upon a time, the Southwest Conference was where football ruled and renegades roamed. It was a land of envelopes stuffed with cash, sham jobs and recruiters bearing more gifts than Santa Claus. At one point, six schools were on N.C.A.A. probation, Southern Methodist football was suspended for a year, and the string of scandals ensnared even a governor of Texas and a Texas A&M trustee who was a part owner of the Dallas Cowboys.

The league was also where nobody paid much attention to basketball.

The Southwest is long gone, a victim of its own excesses and the musical chairs of conference realignment, its members scattered across the college landscape. But for those who might carry some nostalgia for those wild (south)west days, this year’s Final Four is going to pull at the heartstrings the way boosters used to pull out wads of cash.

Two camps of the diaspora will come together in one national semifinal on Saturday after Houston ended a remarkable run by the No. 12 seed Oregon State with a 67-61 victory in the Midwest regional final on Monday. The No. 2-seeded Cougars will play an old colleague, Baylor. The Bears, now in the Big 12, dispatched another Southwest alum, Arkansas, now in the Southeastern Conference, to win the South regional final, 81-72.

Baylor, a top seed, jumped to an early 18-point lead on its deadeye 3-point shooting and fended off the resilient Razorbacks down the stretch, clinching its first Final Four appearance since 1950.

The Cougars, who are in the Final Four for the time since 1984, had a more stressful time getting there. They blew a 17-point halftime lead but Quentin Grimes steadied them, hitting a 3-pointer from the top of the arc with 3 minutes, 21 seconds left, putting Houston ahead to stay.

“One thing you have to remind people, young men in these situations — don’t be afraid to fail,” Coach Kelvin Sampson said, recalling what he told his team during a timeout after Oregon State had tied the score. “Don’t be afraid to miss the shot. You certainly can’t be afraid to take it.”

The Cougars (28-3), who rallied from a 10-point, second-half deficit to beat No. 10-seeded Rutgers in the second round, have navigated a wrecked bracket, facing the 15th, 10th, 11th and 12th seeds. Their latest opponent, Oregon State (20-13), was trying to become the lowest seed to reach the Final Four.

Houston has a rich basketball history, dating to the 1960s when Elvin Hayes fueled a team that threatened U.C.L.A.’s dynasty and carrying through to the mid-1980s, when the high-flying Phi Slama Jama teams — led by the future Hall of Famers Hakeem Olajuwon and Clyde Drexler — went to three consecutive Final Fours, losing back-to-back title games.

Much has happened since then. Little of it has been good for the Cougars.

Relegated to Conference USA once the Southwest disbanded in 1996 and then moving to the American Athletic Conference, Houston had little success fulfilling its athletic ambitions to rival big brother schools like Texas and Texas A&M. It hired Drexler as coach, but he flopped and left after two seasons.

One swing, though, has paid off — hiring Sampson, a sage but scandal-ridden coach. He drew the financial backing of Tilman Fertitta, the restaurant and casino mogul who owns the Houston Rockets. He poured $20 million into a campaign to renovate the Cougars’ Hofheinz Pavilion, now known as the Fertitta Center.

Sampson, who in 2018 led Houston to its first tournament-game victory in 34 years, now has the Cougars in position to win their first championship — and he would do it as an outsider.

The only team from outside the Power 6 conferences to win a championship since U.N.L.V. in 1990 was Connecticut in 2014, after its first season in the American after spending decades in the Big East. Houston fits that profile well with a band of transfers and overlooked talent.

Six Cougars, including four starters, began their careers elsewhere — none more prominently than Grimes, an acclaimed recruit who marked his arrival in college basketball at Kansas by scoring a game-high 21 points in a win over a Michigan State team that ended up in the Final Four.

But Grimes, not as comfortable as he wanted to be, returned home to Houston.

Sampson has been able to pitch a track record of development, burnished by six years he spent as an N.B.A. assistant, though that stint was essentially an exile after he landed Oklahoma and then Indiana on probation for violating N.C.A.A. rules. If this is a redemptive turn, it comes an hour’s drive north of where he was fired more than a decade ago.

If Sampson has rarely had elite talent, he has consistently built winners around hard-nosed ballplayers who build a fortress around the basket they are defending. In his 27 previous seasons as a head coach — at Washington State, Oklahoma, Indiana and Houston — that had been good enough to regularly reach the N.C.A.A. tournament but for just one trip to the Final Four, in 2002 with Oklahoma. The backcourt pair from that team — Hollis Price and Quannas White — are on his staff with the Cougars.

They were surely nodding with approval on Monday night as the Cougars took space away from Oregon State’s floor full of shooters, leaving them flustered. Price and White had to be particularly pleased with DeJon Jarreau, a sinewy 6-foot-5 guard who, like them, is from New Orleans.

A transfer from Massachusetts who spent a year at a community college, Jarreau had 10 points, 8 assists and 8 rebounds, but his best work may have been on defense. He was named the tournament’s most outstanding player. He badgered Ethan Thompson, who had scored 48 points in the previous two games and had been particularly adept at getting to the free-throw line, where he had made 25 of 26 shots in the tournament. On Monday, Thompson was limited to 11 points, though he contributed 7 rebounds and 6 assists to fuel Oregon State’s late charge.

“Man, it’s like a dream come true,” said Jarreau, who is at his third school with his best friend, reserve forward Brison Gresham. “Growing up, watching the Final Four, watching college basketball, you always think about getting to this point, seeing former players on TV do this. Like I said, as a kid, I’m watching TV growing up, and I’m like I hope I’ll be here one day, and man, I’m really here.”

There were similar sentiments when Baylor clinched its berth just after midnight.

As Kool and the Gang’s “Celebration” blared, players bounced around the court, hugged and mugged for selfies, and eventually took turns snipping the nets from the rims.

But in this pandemic season, it was a markedly different celebration. There were only several hundred Baylor fans there to share in the merrymaking, and the traditional confetti shower was a do-it-yourself exercise. The Bears grabbed handfuls from a bucket and tossed them in the air.

As the confetti fluttered down, a portion of the court recalled those days in the Southwest Conference, for it was awash in green.