BATON ROUGE, La. — When the quarterback Joe Burrow visited Louisiana State in the spring of 2018 as he considered where to transfer from Ohio State, he came well prepared. His mother, Robin, met with an academic adviser to map out a graduate studies program, while Burrow visited with the Tigers coach, Ed Orgeron, and two offensive assistants. Burrow brought along his father, Jimmy, a 30-year college coach, and an older brother, Dan, who like their father played at Nebraska.

Before long, Joe Burrow was in front of a board, mapping out protection schemes.

The meeting, scheduled for 30 minutes, lasted two hours.

The main concern on Burrow’s mind: Though L.S.U. had the pedigree to command the spotlight and the talent to breed expectations, it had for years been an offensive backwater, a place where championship ambitions often died.

What assured Burrow that Orgeron, who cut his chops as a defensive line coach, could evolve from the run and play defense formula that Nick Saban and Les Miles used to win national championships more than a decade ago?

“They convinced us that it was going to be different and it wasn’t just going to be overnight,” said Jimmy Burrow, who retired last season as the defensive coordinator at Ohio University. “We asked some pretty pointed questions to make sure this was the direction they were going. You just had to trust them.”

Now, less than two years later, L.S.U. is back in the national championship conversation in a convention-shattering way — with Burrow commanding the most prolific offense in the country.

It was on display Saturday night in L.S.U.’s 42-28 victory over previously unbeaten Florida, as the Tigers eviscerated what had been one of the best defenses in the country, running up 511 yards on a mere 48 plays.

“That’s what all the guys were talking about — we’ve finally got an offense,” said Kelvin Sheppard, a linebacker who had spent the last eight seasons in the N.F.L., as he stood on the sidelines Saturday night with several of his former teammates. “It’s the scheme. Think about if you put Odell and Jarvis in this.”

That would be Odell Beckham Jr. and Jarvis Landry, two of the best receivers in the N.F.L., who were part of an offense — along with other solid pros, receiver Reuben Randle and running back Spencer Ware — that was so predictable it was shut out by Alabama in the national championship game for the 2011 season.

This year’s offense leads the nation with 52.5 points per game and is second behind Oklahoma with 521 yards per game.

The man at the center of the metamorphosis is Joe Brady, a 30-year-old former receiver at William & Mary who was poached from the New Orleans Saints staff to implement a modern run-pass option offense. Brady’s hiring as the passing game coordinator is a sign of how willing Orgeron, the 58-year-old, barrel-chested, gravel-voiced Cajun, has been to evolve.

Another sign came late Saturday night after Derek Stingley Jr., a freshman cornerback who had been repeatedly picked on by Florida quarterback Kyle Trask, made a diving interception in the end zone at the towering, closed end of Tiger Stadium, which held the roars of 102,321 fans like a box canyon.

The turnover gave L.S.U. the ball with 7:26 to play and a 35-28 lead. A commercial timeout provided both teams a chance to gather themselves. It also gave the crowd a moment to take stock of a riveting game, with the then-No.7 Gators (6-1) keeping pace with L.S.U.’s offense by putting together one clock-bleeding drive after another.

L.S.U.’s previous iterations would set about grinding away time, leaving whatever was left on the clock to the defense to kill off.

But as Burrow stood on the sideline getting instructions from his offensive coaches, the rest of the offense huddled around Orgeron on the field. “I told them to look around — this is what we’re here for,” Orgeron said. “We’re built for this. A championship team is going to go down there and score. I challenged them.”

Orgeron and Burrow bumped fists as the quarterback joined the huddle. He then went to work, picking up two quick first downs.

Then, with running back Clyde Edwards-Helaire lined up outside and receiver Ja’Marr Chase positioned in the right slot, Edwards-Helaire broke to the inside, dragging both defenders with him. Chase cut behind and sprinted up the sideline alone, wide open from here to Gross Tete, and Burrow hit him in stride. Chase outraced the safety for a 54-yard touchdown.

“I always want the ball in my hands with the game on the line, and my coaches know that, my teammates know that,” said Burrow, who threw a late touchdown pass to seal L.S.U.’s other marquee victory, on the road at then-No. 9 Texas. “And I think they trust me.”

They are not the only ones. There was skepticism here about the impact a quarterback might have when Burrow arrived before last season, but the locals have come around. One indication is how some are beginning to spell his name: Burreaux.

There is also talk of a Heisman Trophy and his ascending N.F.L. draft stock, bolstered by a performance in which Burrow went 22 of 25 for 293 yards and three touchdowns and didn’t throw his second incompletion until just over a minute remained in the third quarter.

The questions from that recruiting visit have all been answered.

“He wanted to have a chance to win a national championship and we’ve got a shot,” said Jimmy Burrow, one of about a dozen family members who met with the quarterback on his way out of the stadium. “Games like this — that’s why he came to L.S.U.”

And there will be more. After a trip to Mississippi State next week, Auburn will visit Baton Rouge, and two weeks after that L.S.U. will go to meet its perpetual boogeyman: Alabama.

Even if there are still things for the Tigers to clean up — “don’t let good enough get in the way of greatness,” Burrow told his team afterward — they have not been ranked this high since 2012.

This season, L.S.U. is chasing a championship because of its offense — not in spite of it.