Hellebuyck passed Tom Barrasso (Pittsburgh Penguins, 1992-93), who played before shootouts were introduced in the NHL. Hellebuyck had four shootout wins this season.
"I'm very excited about it," said Hellebuyck, who made 30 saves. "To be in the history books is something that I'm always going to look back and be proud of. The fact that this locker room has come together, playing the way we're playing, now the city is behind us the way they are, it's a fun time being in Winnipeg."
WATCH: [All Blackhawks vs. Jets highlights]
Andrew Copp scored twice, and Dustin Byfuglien and Kyle Connor scored for the Jets (52-20-10), who finished second in the Central Division and will play the third-place Minnesota Wild in the Western Conference First Round of the Stanley Cup Playoffs.
Hellebuyck also tied the NHL record for home wins in a season with 30 (Wayne Stephenson, Philadelphia Flyers, 1975-76).
Brent Seabrook scored, and Jeff Glass made 35 saves for the Blackhawks (33-39-10), who are eliminated from playoff contention. This was the final regular-season game for each team.
"It [stinks]," Seabrook said. "I've said a million times, I think it's the best part of the season coming up here. To not be a part of it, not playing any meaningful games for us down the stretch, it's been tough."
Byfuglien gave the Jets a 1-0 lead at 15:12 of the first period on a 5-on-3 power play. His point shot beat Glass high to the blocker side.