Paul Lasike run for a loss of 3 yards to the BYU 35 Paul Lasike fumbled, forced by Jackson Dillon, recovered by BYU Adam Hine (0:50)

MIAMI -- They battled for 60 minutes, then again for two overtimes and literally fought after the final play.

And for Memphis, the end result was a win 76 years in the making.

A midfield brawl between Memphis and BYU broke out -- with many players from each team punching and grabbing -- after the final play of the game. Rob Foldy/Getty Images

Paxton Lynch threw four touchdown passes and rushed for three more scores, Jake Elliott kicked a 54-yard field goal to end the first extra session and Memphis wasted a pair of double-digit leads before rallying to beat BYU 55-48 in the inaugural Miami Beach Bowl on Monday.

"It wasn't always pretty," Memphis coach Justin Fuente said.

No, but it was memorable -- largely for the right reasons, though also for a scene that got out of hand at the end.

Roderick Proctor caught an 11-yard pass from Lynch to put Memphis up in the second overtime, and from there things just got worse for BYU.

After DaShaughn Terry sealed the win by picking off BYU quarterback Christian Stewart in the second OT, tensions and emotions took over. Dozens of people from both sidelines spilled toward the middle of the field, many punching and grabbing. Cameras caught BYU defensive back Kai Nacua -- who had blood streaming from his face -- coming from behind to punch in the head Memphis tight end Alan Cross, who was being restrained by someone from the Tigers' staff.

"You want them to rise above that and have fantastic sportsmanship," BYU coach Bronco Mendenhall said. "I'm sure if you go back and look, there'll be an instance or two that ignited. Probably the majority wanted to handle it really well."

BYU athletic director Tom Holmoe apologized to fans via Twitter.

We expect better of our athletes, even in the face of a difficult loss. We intend to fully review this matter. I apologize to Cougar Nation. — Tom Holmoe (@TomHolmoe) December 23, 2014

Lynch threw for 306 yards and the four touchdowns, including a 5-yarder to Keiwone Malone with 45 seconds left in regulation -- on fourth-and-4, his second fourth-down conversion on the Tigers' final drive of regulation. And Lynch said afterward that Malone wasn't even the intended receiver.

"This is probably the craziest game, most emotional game I've ever been a part of," said Lynch, the first player at the FBS level with at least four TD throws and three TD runs in a game since Clemson's Tajh Boyd in 2012.

Malone caught two TD passes for the Tigers (10-3), who won their first bowl game since 2005 and reached double-digits in wins for the first time since 1938. The 10 wins this season match Memphis' total from the previous four seasons combined.

"We made history," Malone said.

Stewart passed for 348 yards and three touchdowns for BYU (8-5), which finished with the same record for the third straight season and fell to 0-7 all-time in the state of Florida.

The Cougars rallied from 10-point deficits twice to take the lead, but couldn't hold either. BYU has now lost 12 straight games in which it trailed at some point by double figures.

"I was sitting there in the shower replaying in my mind just all of the missed opportunities with guys open or just missed opportunities that we had," Stewart said. "We played a good enough game to win this. I feel I missed a few key throws that really could have sealed the deal for us."

The teams combined for four touchdowns in the first 8 minutes and Memphis led 24-14 early in the second quarter, before BYU rallied with two scores in the final 4:55 to take a 28-24 lead at intermission.

"Nothing was conservative, from the start," Mendenhall said.

Memphis opened the second half with a 13-play, 71-yard march capped by Lynch's third rushing score of the day to reclaim the lead, and the Tigers extended the edge to 38-28 after Cross caught a 17-yard pass from Lynch with 3:45 left in the third.

But back came BYU again, with a field goal from Trevor Samson and -- after the Cougars stripped the ensuing return away from Memphis' Joe Craig for a fumble -- Paul Lasike's second touchdown tied it at 38-all with 10:52 left.

Zac Stout's interception return for a score that put BYU on top came 3 minutes later. The Tigers, though, had the final answer.

"I don't know what happened at the end, first of all, so I don't know how upset I am," Fuente said. "I'm sure I'll see it eventually. It's not who we are, it's not what we want to represent. I hope it doesn't take away from an incredible football game, for both sides."