Jonathan Toews also scored during a man advantage, ending a seven-game goal-scoring drought, as the Blackhawks won for the third time in four contests and beat the Jets in Chicago for the second time this week.
Kane, the NHL's leading scorer, got his 19th goal midway through the second period to match Sidney Crosby's 25-game streak, set in 2010-11, for the longest in the league since 1992-93. Kane has 16 goals and 23 assists during his run.
''Obviously, I feel honored and very fortunate to be involved with names like those,'' Kane said. ''It's something I'm trying not to think about it too much. Just go out and play the game.
''The last few games, I don't think I've played as well as I probably could have, but sometimes you end up with those chances and you can take advantage and get a point.''
Kane's goal made it 2-0 as Chicago went 2 for 5 on the power play, and killed all five Winnipeg advantages one night after a lackluster 5-1 loss in Nashville.
''It was a different type of game out there,'' Kane said. ''A lot of penalties both ways. Obviously, we cashed in on a couple power-play goals; that was the difference in the game.''
Crawford wasn't heavily tested in recording the 15th shutout of his career as the Blackhawks blocked a season-high 23 shots in front of him.
''Our penalty kill did a great job,'' coach Joel Quenneville said. ''They were disruptive in the neutral zone. They were excellent in the defensive zone.''
''I liked his game tonight, boy he was good - solid,'' Jets coach Paul Maurice said. ''I'm not looking at him on either one of (the goals).''
What Maurice did look at: Chicago had 10 shots on its five power plays, while limiting the Jets to only one shot on their five advantages.
''That's the difference in the game,'' Maurice said.
The Jets also played for the second straight night after beating Columbus 6-4 at home on Thursday.
The two teams met at the United Center for second time in five nights. Chicago beat Winnipeg 3-1 and goalie Michael Hutchinson on Sunday.
The Blackhawks rebounded from their loss in Nashville with an energetic first period, outshooting the Jets 13-7 and taking a 1-0 lead.
Toews connected on a power-play 4:34 in, wrapping in a loose puck after Hellebuyck stopped Andrew Shaws' deflection of a shot from the point and Marian Hossa's point-blank rebound attempt.
Kane made it 2-0 at 10:36 the second, completing a crisp give-and-go with Teuvo Teravainen across the slot. Kane fired a rising one-timer from the left circle that beat Hellebuyck over the shoulder for his 10th power-play goal.
''What a play. What a shot,'' Quenneville said. ''It's fun being a part of it, fun watching him continue to find different ways.''
Crawford had to be sharp during a couple of flurries in the third and made a point-blank stop on former Blackhawk Andrew Ladd midway through the period.
NOTES: Chicago D Trevor Daley returned to the lineup after missing two games with what he called a neck injury, suffered on Sunday in a collision against the Jets. ... The Blackhawks assigned F Marko Dano to Rockford of the AHL earlier on Friday. . Chicago D Michal Rozsival was healthy scratch for the first time since coming back from a broken ankle on Nov. 14.