Laine, picked second by the Jets in June, decided the game seconds after Matthews, the first pick by the Maple Leafs, was stopped on a breakaway. That brought the Jets all the way back from trailing 4-0.
If there's more of this to come, the NHL and fans in Winnipeg and Toronto are in for a great ride.
"I can't speak for their room, but it's not as big a story in our room going into the game," Jets coach Paul Maurice said. "And then it just becomes that with the last play of the game. Two great young players that are going to give their fans reasons to have nights like this over the course of the year and their careers.
"Starting now. We've just seen brilliance from both of them in the first four games of the season. Lucky for Winnipeg fans and lucky for Toronto fans, they'll get to watch two great young players."
Matthews scored four goals in his first NHL game, the first player to do that in the modern era (since 1942-43).
Video: TOR@WPG: Laine erupts for first NHL hat trick
The 19-year-old had an assist in the second period against the Jets, a clever pass to linemate William Nylander for a power-play goal that put Toronto ahead 3-0.
Laine, with his three goals, shares the NHL lead of four with Matthews, David Pastrnak of the Boston Bruins, Richard Panik of the Chicago Blackhawks and Vladimir Tarasenko of the St. Louis Blues.
The end of this was not unlike one on March 1, when the top two picks of the 2015 NHL Draft went head-to-head in an overtime finish.
In that one, Jack Eichel, the second pick by the Buffalo Sabres, narrowly missed a backhand shot on an overtime drive to the net. When the play went back up ice, Connor McDavid, the first pick by the Edmonton Oilers, scored the game-winning goal.
On Wednesday, Jets goalie Michael Hutchinson stopped Matthews' low shot on the overtime breakaway attempt. Winnipeg defenseman Dustin Byfuglien retrieved the puck and sent it quickly up to Laine, who went on a 2-on-1 with Nikolaj Ehlers.
Laine didn't even look at his teammate and snapped a hard shot under the crossbar to the glove side of Toronto goalie Frederik Andersen.
"With his shot, I don't think he should be looking," Jets center Mark Scheifele said. "He's a special player. He gets better and better every night. And that's what you saw tonight."
Video: TOR@WPG: Laine nets hat trick with GWG in overtime
Laine's first goal came one minute into the third period. He stopped a hard Byfuglien pass near the net then eluded Toronto defenseman Martin Marincin and in the same motion sent a quick wrist shot high into the net to make it 4-2.
With Hutchinson at the bench for an extra attacker in the final minute of the third period, the puck came to Ehlers at the point. He faked a slap shot and passed across to Laine, who one-timed a shot into the open side with 54.5 seconds remaining.
"I knew if put it over there, he was going to put it in," Ehlers said. "Because I know how he can shoot the puck. If he gets it there, he can put it in."
Laine, 18, barely cracked a smile after the game.
"I think it was just a normal regular-season game," he said. "Those two periods we were playing so bad, I don't know why, including me. You shouldn't let the other guys score four goals and then start to play.
"The third period was just amazing from us, again."
Laine said he hit his target with his overtime chance.
Video: WPG@TOR: Laine fires a PPG, ties game late in the 3rd
"I noticed that we were two against one with Nicky and I'm better on two against one from the other side, so I decided to shoot and it just went right where I wanted it to be," he said. "It was an amazing feeling."
Matthews said his overtime chance, one of his seven shots on goal, was not his best.
"It's late in the game in overtime," he said. "It's tough to make moves, the ice is a little rough, so I tried [to] slide it in there. The goalie read it well, and it's 3-on-3, so you get the puck going one way, it's an odd-man rush the other.
"I think in the first place, we shouldn't have been in overtime … we'll fix something and move forward to the game tomorrow [at the Minnesota Wild]."
Laine said he heard the pregame talk about the rivalry between him and Matthews.
"He's a good player, so it's nice to play against him," Laine said. "I was in the same draft as him, so people are comparing us to each other. I know that. But our team won (against) their team. It was a good game, so I'm happy."
Laine said he also heard the boisterous crowd at MTS Centre on Wednesday.
Video: TOR@WPG: Laine rips a wrister past Andersen
"Yeah, who wouldn't hear them?" he said. "It's just an amazing home crowd and they're supporting all the time when we were chasing 4-0. It's just amazing to play here and they're just amazing fans and it's good that they're going to support us no matter what."
He said he also heard the Winnipeg faithful during the third period, chanting, "Laine's better!"
"Yeah, I heard something and that sounds pretty good," he said. "Yeah, that's their opinion and I'm not going to say anything about that."
Matthews said he heard the chant too.
"It's a great atmosphere," he said. "Obviously, Winnipeg is a great hockey city, so that's what you get when you play these types of teams. You [have] to block it out and just have fun with it."
Maurice said he now understands something he did not in the preseason.
"I've never seen anybody score goals like Patrik Laine in practice, ever, in 20 years," Maurice said. "He has a gift, no question about it. It's a different shot. He has his eyes up and he has the hands for it.
"There were shots in exhibition off posts and near misses and he was really frustrated. I remember thinking that he's probably getting too frustrated with that.
"Now I realize why he was so frustrated with that. Because he's used to them going in."