Hagelin scored to make it 2-1 at 14:32 of the second period, receiving a lofted pass from Nic Dowd upon exiting the penalty box and beating Hellebuyck five-hole on a breakaway.
"I think you always have a dream that it might land on your stick, and today was one of those days," Hagelin said. "I was going to come in and help our team defensively, and then [Dowd] got the puck, so I might as well take off and try to go for the breakaway and he threw me a nice alley-oop."
Winnipeg outshot Washington 26-13 in the final two periods.
"Spent most of the game down there, on the power play a lot, we just couldn't finish," Jets captain Blake Wheeler said.
Winnipeg appeared to tie it at 9:43 of the third period, but the referee ruled no goal because of goaltender interference on Bryan Little. Maurice challenged the play, but the call was upheld after a video review.
"I saw a loose puck kind of between his legs and I didn't hear a whistle, so I was going for it," Little said.
Eller added an empty-net goal with 22 seconds remaining to extend the lead to 3-1.