Leo Carlsson and Radko Gudas also scored, the latter pulling the Ducks even under a minute after the Jets had claimed a 3-2 lead with less than three minutes to play in regulation. Ryan Strome, Alex Killorn, Brian Dumoulin and Frank Vatrano collected assists, as did goaltender John Gibson, who earned his sixth win of the season with 27 saves.
Mark Scheifele, Alex Iafallo and Adam Lowry scored for Winnipeg. Hellebuyck, a two-time Vezina Trophy winner, took just his sixth loss in 31 starts this season.
Anaheim had plenty of scoring chances early in a back and forth opening period, but it would be Winnipeg jumping out to a 2-0 lead on a pair of chances in tight against Gibson.
Scheifele put Winnipeg ahead on the game's first shift, capitalizing on a Ducks turnover via a centering pass by linemate Kyle Connor.
The goal, the 320th of Scheifele's NHL career, moved the longtime Jet within eight of Ilya Kovalchuk's franchise record. This season, Scheifele is tied for second among all NHL players with 23 goals in 40 games played.
Lowry doubled the lead late in the period on a 2-on-1 rush alongside Iafallo, converting a terrific backdoor saucer pass from the former King.
Lowry, the Winnipeg captain, became the sixth different Jet to reach double-digit goals this season, trailing only Washington and Colorado for most in the NHL.
Anaheim would get back within one before intermission though as Carlsson found open space in the slot and redirected a Gudas point shot off the post and then off Hellebuyck before it trickled just over the line.