Connor Hellebuyck made 25 saves for his 36th win.
Brandon Saad and Patrick Sharp scored for the Blackhawks (30-33-8), who are 15 points behind the Dallas Stars for the second wild card into the Stanley Cup Playoffs from the Western Conference.
Anton Forsberg allowed three goals on six shots and was pulled at 9:26 of the first period. J-F Berube made 30 saves in relief.
"Consistency's been our biggest challenge all year, in games or game to game," Blackhawks coach Joel Quenneville said. "You get a break like that right off the bat and you got some excitement in your game, we had a heck of a game in here the last time, and then it's like, 'OK, you're down 5-1 after one, that's the ballgame right there.'"
Laine, who turns 20 on April 19, has 16 goals and eight assists during his streak after he helped set up Paul Stastny's goal to tie the game 1-1 at 3:43 of the first period. He passed Colorado Avalanche forward Nathan MacKinnon, who had 18 points (five goals, 13 assists) in a 13-game streak as an 18-year-old from Jan. 25-March 6, 2014.
"It sounds pretty good," Laine said. "I've been working super hard lately, and I've been getting rewarded, offensively, these past couple of weeks. But it sounds good to me."
Saad scored 11 seconds into the game to give Chicago a 1-0 lead. Winnipeg coach Paul Maurice challenged for offside, but the goal was upheld after video review.