Hellebuyck's shutout was his second this season and 30th in the NHL.
"Our details have been right," Hellebuyck said. "We're getting everyone's 'A' game right now. We've got a lot of character in this room and it's starting to show. We care about winning and we're showing it."
Pierre-Luc Dubois and Josh Morrissey each had a goal and an assist for the Jets (7-3-1), who are 5-0-1 in their past six games. All four goals came on special teams (three on power play, one short-handed).
Dylan Wells made 12 saves in his NHL debut for the Blackhawks (5-5-2). He entered the game at the start of the third period after Arvid Soderblom (21 saves) left because of an undisclosed injury.
"That's been a goal of mine since I was a kid, so I'm really happy," Wells said. "At the end of the day, my job is to stop the puck, it doesn't matter what league that's in, whether that's here or in the AHL. When I got told I was going in, it was just, 'Keep the mindset simple, stop the puck.'"