GLENDALE, Ariz. — Gonzaga’s seemingly Sisyphean quest to reach the Final Four is over. College basketball’s perennial bridesmaid with the funny name finally has a Final Four engagement, and it has the best team of the quartet competing for the shiniest moment of them all here in the desert.

If Duke, North Carolina, or Kansas rolled into the Final Four as a No. 1 seed with a 36-1 record, the nation’s top-rated defense, a 22.3-point average margin of victory, a deep and talented roster replete with upperclassmen and a potential NBA first-round pick freshman, and the Associated Press Coach of the Year, those schools would be the clear favorite to cut down the nets. Gonzaga has gone from the little engine that could to a roaring jet turbine. But the nouveau-riche Zags still don’t quite get the same level of respect of their more deep-rooted college basketball brethren. They can’t quite outrun their West Coast Conference roots.

That should make the Zags sentimental and actual favorites here in Arizona, where they’re joined by South Carolina, Oregon, and North Carolina. It’s an odd collection of teams that hopefully produces an outstanding denouement to a blase NCAA Tournament. (A Florida buzzer-beater and one UNC unsung hero do not a tournament make.) Gonzaga isn’t content to settle for filling the blank space next to Final Four on their college basketball upper-crust club application. The Bulldogs are focused on winning it all.

To do that they’ll have to get through the dogged South Carolina Gamecocks, who won the East region as a No. 7 seed. The Gamecocks have March Magic and scintillating guard Sindarius Thornwell on their side. Gonzaga vs. South Carolina is the first matchup on Saturday at University of Phoenix Stadium. It’s a defensive duel. The Zags rank No. 1 in the nation in defensive efficiency, allowing 85.8 points per 100 possessions, according to college basketball reference.com. The Gamecocks rank fourth at 91.1 points allowed per 100 possessions.

North Carolina, which lost in the title game last year to Villanova, takes on Oregon in the nightcap.

It shouldn’t take a Final Four or a national championship to recognize the incredible job Gonzaga coach Mark Few has done in raising the Spokane, Wash., program to national prominence. Gonzaga long ago ceased being the answer to a trivia question — where did NBA Hall of Famer John Stockton go to college? — and became one of the most consistent programs in the country.

Gonzaga has made the tournament 19 consecutive years, a streak topped only by Kansas, Duke, and North Carolina, and been to eight Sweet 16s since 1999.

“In terms of getting respect, that all depends on who you ask,’’ said Gonzaga guard Jordan Mathews. “There are some people that will say, ‘Oh yeah, Gonzaga is right up there with Duke and Carolina.’ There are others who would say, ‘Well, Duke and North Carolina have five championships.’ It’s all a little bit different. It depends on whom you ask. We don’t focus on that. All we can focus on is how hard we play. To be 36-1 is pretty good, and that’s what we cherish and what we hang our hat on.’’

Few, who spent 10 seasons as a Gonzaga assistant, inherited the program from Dan Monson in 2000, after the Zags first burst onto the scene with a Cinderella Elite Eight run in 1999. The unflappable Few is the winningest active coach by winning percentage in Division 1 college basketball with a 466-111 record (.808).

Few always believed the Final Four was just the next step for Gonzaga, not validation of the program.

“Certainly, I felt my stance all along was you just got to be good enough, and then eventually it’s going to happen,’’ said Few. “We wanted to stay nationally relevant, and I think we’ve done that year after year. I think that’s probably what I’m most proud of, and then eventually you kick the door down and break through like we did this year.’’

Gonzaga was certainly good enough this year. The Zags were the No. 1 team in the nation for four weeks and were 29-0 before they lost their regular-season finale to BYU.

In the era of the one-and-done, Gonzaga has plenty of experience. They just didn’t have a lot of experience playing with each other.

Nigel Williams-Goss (Washington), Mathews (Cal), and Johnathan Williams (Missouri) are all transfers. Those players and NBA ear-marked freshman 7-footer Zach Collins had to coalesce with returning players Przemek Karnowski, Josh Perkins, and Silas Melson. Egos and agendas had to fade into the background for Gonzaga to reach the main stage of March.

In September, Gonzaga took a team camping trip across the nearby Idaho border to Hayden Lake, Idaho. The great outdoors sojourn served as a get-to-know-you and a team bonding experience for what was at that time an enviable collection of talent, not a team. It made players heed Few’s call to buy into the Gonzaga culture.

“We truly did buy in,’’ said Williams-Goss, the team’s leading scorer and a 2013 McDonald’s All-American. “We said we’re going to put all of our individual stuff aside. We have all the talent in the world that we need to get to Phoenix and to win a national championship. Now, we need to do our part. We can’t just talk about sacrificing. We can’t just talk about brotherhood. We have to actually go and show it.’’

They have shown it. They’ve also built upon the accomplishments of previous Gonzaga players who fell short, names like Ronny Turiaf, Adam Morrison, Dan Dickau, current Celtic Kelly Olynyk, Kyle Wiltjer, and Domantas Sabonis.

Those Gonzaga hoops alumni have reached out and thanked them for finally putting the program where it belongs.

“It meant the world to me be able to give them that gratification because they’re the ones that laid the foundation. If they didn’t make those NCAA Tournament runs and have all the highlights they had we wouldn’t be sitting here with this jersey on today,’’ said Williams-Goss.

“. . . We’re very happy that we were able to be the first team from Gonzaga to make it to the Final Four, but we feel like a national championship banner deserves to be hung inside The Kennel. We’re going to give it all we have to try to make that happen.’’

The Zags have always been here, but they’ve finally arrived.

Christopher L. Gasper can be reached at cgasper@globe.com. Follow him on Twitter @cgasper.