Laine is the third player since the 1997-98 season to score five goals in a game, the first since Johan Franzen did it for the Detroit Red Wings on Feb. 2, 2011. Marian Gaborik did it for the Minnesota Wild on Dec. 20, 2007.
"I feel like every time I touched the puck, it kind of went in somehow," said Laine, who scored on each of his five shots. "I guess it was one of those nights. It's probably not going to happen in a while, but we'll hope it happens. I think it was a great night overall for myself and my linemates.
"I was just trying to score one in the game. I wasn't thinking about NHL records. ... Obviously it was a great achievement, but it was a great win too."
[WATCH: All Jets vs. Blues highlights]
Laine, who will turn 21 on April 19, has 11 goals during a four-game goal streak and his seventh NHL hat trick is the second most before the age of 21, done in 177 games; he broke a tie with Dale Hawerchuk and Jimmy Carson, who each had six, and trails Wayne Gretzky, who had 12 in 210 games.
Video: WPG@STL: Laine's five-goal game makes Jets history
Laine is the 14th player in NHL history and sixth in the modern era (1943-44) to score at least three hat tricks in the first 22 games of a season.
"It's fun to play," Laine said. "I'm having some great plays with my linemates. It's pretty easy to play hockey right now with those guys (Kyle Connor and Bryan Little). I've just got to keep working hard and keep doing the things we're doing."
Laine, who leads the NHL with 19 goals, scored three goals in the second period.
Blake Wheeler had a goal and an assist, Little and Connor each had four assists, and Laurent Brossoit made 23 saves for the Jets (13-7-2), who ended a four-game road trip 2-2-0 and are 3-0-0 against St. Louis this season.
Laine has 13 goals in 11 games against the Blues, including six this season.
"To be honest with you, I'm surprised it hasn't happened sooner," Jets coach Paul Maurice said of Laine scoring five goals. "I've said that once only because I keep it to myself, but on the nights that he's scored his three, he's had his chances to get so much more. Boy, those guys were so good, and some of those goals, you've just got to sit back and smile if you're standing on the right bench."
Video: Clutch Performance: Laine scores 5 goals in rout
David Perron had a goal and an assist for St. Louis (8-11-3), and Chad Johnson allowed six goals on 25 shots (four to Laine) before being replaced by Jake Allen, who made nine saves.
The Blues were playing the second game of a back-to-back; they defeated the Nashville Predators 6-2 on Friday.
"This is a tough one to swallow, especially after the effort we gave last night, beating a team like that," Blues captain Alex Pietrangelo said. "We've got to be ready to play. It's a back-to-back, early game, we've got to be ready to play."
Laine scored at 16:26 of the first period to give the Jets a 2-1 lead. Vladimir Tarasenko tied it 2-2 at 17:50.
Laine's power-play goal at 1:41 of the second period gave Winnipeg a 3-2 lead, and Brandon Tanev scored to make it 4-2 at 10:50.
Laine scored his third goal to make it 5-2 at 12:53 of the second, and his fourth made it 6-2 at 16:02.
His fifth goal came 1:16 into the third to make it 7-2.
Ryan O'Reilly made it 7-3 at 7:42 of the third, and Pat Maroon scored a power-play goal to make it 7-4 at 14:32.
Brendan Lemieux scored with 40 seconds remaining to make it 8-4.
Perron's first goal in 14 games gave St. Louis a 1-0 lead at 13:42 of the first period. Wheeler tied it 1-1 at 15:45 of the first.
Video: WPG@STL: Wheeler scores on a fortunate bounce
They said it
"Just get the puck to Patty. He was feeling it tonight. It's pretty special. We all know what kind of shot he has." -- Jets forward Kyle Connor
Video: Enterprise Hat Trick: Johnsson, Guentzel, Laine
"It's just his shot. He's got good linemates. Wheeler found him a couple times and he's just wide open. If you give him time and space with that wrist shot, he's going to score the majority of the time. He's got one of the best shots in the League." -- Blues defenseman Joel Edmundson
Need to know
Laine is the first player to score four goals this season and fourth player in Jets/Atlanta Thrashers history to score four. The others were Mathieu Perreault (Jan. 13, 2015 vs. Florida Panthers), Ilya Kovalchuk (Nov. 11, 2005 vs. Tampa Bay Lightning) and Pascal Rheaume (Jan. 19, 2002 vs. Panthers). ... Laine has 99 NHL goals. Only six NHL players have scored 100 before turning 21; Gretzky (167), Carson (141), Hawerchuk (122), Steven Stamkos (114), Kovalchuk (108) and Brian Bellows (102). ... Tarasenko's goal was his first in 11 games. ... Wheeler played his 800th NHL game.
Jets: Host the Pittsburgh Penguins on Tuesday (8 p.m. ET; TSN3, ATTSN-PT, NHL.TV)
Blues: At the Detroit Red Wings on Wednesday (7 p.m. ET; NBCSN)
Video: Laine's five goals makes Jets history in 8-4 win