Little beat Bruins netminder Chad Johnson with a low shot between his pads, the only scorer in three rounds.
Jets forward Evander Kane 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 (36-35-10) to a victory.
Brad Marchand had scored the opening goal at 10:12 of the first period for the NHL-leading Bruins (53-18-9), who have clinched top spot in the Eastern Conference and have 115 points, three ahead of the idle Anaheim Ducks.
Winnipeg veteran forward Olli Jokinen was called for holding with 40 seconds left in overtime, but Jets rookie netminder Michael Hutchinson made a snapping glove save on a blast from David Krejci with seven seconds left to send it to a shootout.
Hutchinson stopped 32 shots in regulation and overtime, while Johnson made 36 saves.
Hutchinson was drafted by the Bruins in the third round (77th overall) in the 2008 entry draft. He was signed by the Jets prior to this season.
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.
The Bruins host Buffalo on Saturday and finish the season Sunday on the road in New Jersey.
Winnipeg ends its regular season on Friday in Calgary.
Marchand opened the scoring when he used Reilly 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.
The Jets outshot the Bruins 12-6.
Boston picked up the shots to 21-20 after a scoreless second period that saw Winnipeg defenceman Mark Stuart throw his weight.
First Stuart smacked into Loui Eriksson standing at the side of the Jets net and knocked him into the backboards. Eriksson wasn't injured and Stuart didn't draw a penalty.
A few minutes later, Stuart sent Marchand to the ice with a high stick. Marchand got up slowly, but returned to action while Stuart was sent to the box.
Winnipeg leading scorer Blake Wheeler set up Kane's tying goal when he raced down the side of the ice and sent the pass to Kane in front of the net.
It was Wheeler's 41st assist of the season and 64th point, which bumped up the career high he's set this season.
The Jets had a 35-31 edge in shots of goal after regulation.
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