Adam Lowry tied things up for Winnipeg at 10:49 of the opening frame as his shot from the faceoff dot beat Corey Crawford up high.
The Jets made it 2-1 with just 34 seconds left in the first, courtesy of Jack Roslovic.
Crawford denied Mark Scheifele's penalty-shot attempt 40 seconds into the second period to keep it a one-goal game.
However, Roslovic scored his second of the night at 5:11 of the second to give the visiting Jets a 3-1 advantage.
Strome put Chicago back within one just moments later as he got in behind the defense and potted his 18th of the season. Drake Caggiula and Gustafsson recorded the helpers.
It looked as though Scheifele closed out the game by giving Winnipeg a 4-2 lead with three minutes remaining in regulation. However, Chicago challenged that the goal was offside and the call was reversed, making it 3-2 once again and giving the Blackhawks a chance.
Strome played the hero role with his second of the night at 17:40 of the third period, tying the game at 3-3. He was assisted by Toews and Gustafsson.
Crawford was magnificent early in overtime, but Hayes managed to beat the Blackhawks netminder for the win.
CAREER YEAR FOR THE CAPTAIN:Toews' goal in the first period gave him 34 for the season, which ties his career-high set in 2008-09. It was also his 77th point of the season, which is now a new career-high for the captain, surpassing the 76 he recorded back in 2010-11. Toews later added an assist to increase his new career-high to 78 points. What a year of offense it has been for Toews and the Blackhawks.
TALKING POINTS:
"It would have been nice to get a win. But it seems like that's a recurring theme. A lot of games just slip away from us and it's tough to make the playoffs when that happens." -Corey Crawford on the loss to Winnipeg
"We're working hard every day...There hasn't been a game where we've felt terribly about ourselves and we're right in every game."- Dylan Strome on the Blackhawks performance down the stretch
