That will happen when it's a 2-1 shootout victory against the NHL-leading Boston Bruins, a team that cast him aside.
Hutchinson, who made 32 saves in only his second NHL game, was drafted by the Bruins in the third round (77th overall) in the 2008 entry draft and was signed by the Jets prior to this season.
"You really want that first win against them, kind of show them what they gave up on," said the 24-year-old.
Bryan Little scored in the shootout in what was Winnipeg's (36-35-10) last home game of the season. They finish the schedule Friday in Calgary.
Little beat Bruins netminder Chad Johnson with a low shot between his pads, the only scorer in three rounds.
Olli Jokinen missed for Winnipeg, while Reilly Smith, Ryan Spooner and Brad Marchand missed for the Bruins.
Marchand did score Boston's only goal of the game in the first period.
Jets forward Evander Kane took a pass from leading scorer Blake Wheeler and tied the game 1-1 with a wrist shot that beat Johnson over his glove with 1:57 left in the third period.
Just a couple of minutes earlier, Kane had hit the post after fans started chanting "Go Jets Go" to try to propel their team to a victory.
Jokinen was called for holding with 40 seconds left in overtime, but Hutchinson made a snapping glove save on a blast from David Krejci with seven seconds left to send it to a shootout.
"He was outstanding," Wheeler said of Hutchinson, whose first NHL game was a 1-0 loss on Monday against Minnesota.
"He really kept us in the game. They had some point-blank opportunities to add to their lead and he kind of put us away and he gave us an opportunity to tie the game."
Johnson made 36 saves and Marchand scored his 24th goal of the season at 10:12 of the first period for the Bruins (53-18-9).
While Boston has clinched top spot in the Eastern Conference and have 115 points, three ahead of the idle Anaheim Ducks, they have just one win in their last five games (1-1-3).
Head coach Claude Julien was none too happy with the latest effort.
"We looked like a disinterested team tonight," Julien said.
"(The Jets) were determined to have a good, strong finish here and they did. They were the better team tonight from start to finish.
"I thought our goaltender was the reason we were able to come out of here with at least a point."
The fans gave the Jets a standing ovation after the game, and the players skated to centre ice and raised their sticks toward the sold-out crowd of 15,004 at MTS Centre.
It was the third straight season the local fans bid farewell to the Jets without watching a playoff game, and the seventh season in a row the former Atlanta franchise has missed the post-season.
"They show up every night and cheer us on hard and we're very appreciative of that," Wheeler said.
"The results aren't where we want them to be so the only thing we can do is try to work on that and improve in the future."
Wheeler recorded his 41st assist of the season and 64th point, which bumped up the career-high points he's set this season.
The Bruins host Buffalo on Saturday and finish the season Sunday on the road in New Jersey.
Marchand agreed with his coach that the team wasn't fine-tuning itself for the playoffs.
"It felt more like an exhibition game out there," Marchand said.
"The only thing that would have impressed was 1-0 till the last minute. We can play much better hockey than that."
Marchand opened the scoring when he used Smith's dumped-in shot off the back boards to beat Hutchinson with a backhand shot.
Jarome Iginla almost added to the lead with a breakaway, but Hutchinson came out and stopped the wrist shot with just under four minutes to go in the period.
Winnipeg had a 35-31 edge in shots of goal after regulation.
"He read those seams perfectly," Jets head coach Paul Maurice said of Hutchinson, who will be returned to the St. John's IceCaps for their AHL playoffs.
"His push was perfect, square on that shot, and he knew where that was going. The players played as hard as they could to give him a chance to be great and win and that's what happened. He was great, we win."
Notes: Boston rested centre Patrice Bergeron and defenceman Zdeno Chara. … Winnipeg was without captain Andrew Ladd (elbow) and defenceman Dustin Byfuglin (upper-body), who won't play the final game.