Dmitry Kulikov, Kyle Connor and Mark Scheifele scored for Winnipeg, which killed off three penalties and is 16 of 16 on the penalty kill in its last 10 games.
A big reason has been Hellebuyck, who tied Columbus' Elvis Merzlikins and Vegas' Marc-Andre Fleury for the NHL lead in shutouts.
''I love shutouts,'' Hellebuyck said. ''Tonight, great team shutout. I definitely love that on the resume, and it's fun seeing it.''
Hellebuyck robbed Alex Ovechkin at least twice and made a great glove grab off recently acquired Ilya Kovalchuk during a second-period power play. He also stopped forward Jakub Vrana on a breakaway early in the third period.
''The guys were battling really hard in front of me and were playing extremely well,'' Hellebuyck said.
Ilya Samsonov made 30 saves for Washington two days after the Capitals won the first game of the home-and-home series 4-3 in a shootout.
''We know we can be better,'' Capitals forward Tom Wilson said. ''And even when there's a good goalie in there, not scoring any goals isn't going to get the job done. You've got to find a way to get in front of him or make them more difficult or bear down on our chances. It happens.''
Kulikov beat Samsonov with a screened shot from the blue line 6:33 into the game. Blake Wheeler and Neil Pionk assisted on the eventual winner.
After a scoreless second period, Connor gave the Jets a 2-0 lead 7:20 into the third. He converted a pass from behind the net from Scheifele, who extended his point streak to six games. It was Connor's 32nd goal of the season and 99th of his career.
Scheifele added an empty-net goal on an assist from Connor with 1:28 left to play.
''He was fantastic,'' Scheifele said of Hellebuyck. ''He's been fantastic for us all year. For him to get the shutout against a team like that is pretty special. It was a big win for our team.''
Winnipeg lost forward Patrik Laine early in the first after he blocked Michal Kempny's shot, which hit his left foot. It was Laine's first shift of the game and lasted just 23 seconds. Jets coach Paul Maurice said tests had already been done and nothing was broken.
''Every guy's got to make a sacrifice,'' Hellebuyck said. ''Obviously you never want to see anyone not return to the game. In this playoff chase, you've got to take a hit, you've got to make a hit, you've got block a shot. Do the right things all over. Obviously, it (stinks). Hopefully, we get some good news.''
Washington coach Todd Reirden said Winnipeg losing Laine meant the Capitals saw more of the Jets' top line, but also credited Hellebuyck.
''I thought their goalie played well,'' Reirden said. ''That being said, I thought we could have made it more difficult with a little more net traffic and a little more net presence and delivered a few more pucks, and when we had the opportunity, make him go laterally.
''Some of the chances we had, we had some open nets and some breakaways and some opportunities and he made some good saves. So, sometimes, you have to give him credit. But I would have liked our chances to be at a higher number than it was tonight.''
NOTES: The Jets are 6-2-1 in their last nine home games. ... Jets D Josh Morrissey missed his third straight game with an upper-body injury.
Capitals: at Minnesota on Sunday night.
Jets: at Edmonton on Saturday before returning home for a three-game homestand next week.
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