Michal Handzus, with assists from Brandon Saad and Patrick Kane, scored his second goal of the play-offs two minutes into the second extra frame to seal the dramatic win and cut the series deficit to 3-2.
"It's a huge win for us and it's always nice to contribute," said Kane, who had just a single point in the series before Wednesday's game, when he had four assists.
Chicago gained an early lead with goals from defensemen Brent Seabrook and Johnny Oduya in the first four minutes of the match.
The Kings' Jarret Stoll briefly narrowed the lead midway through the period before Saad recaptured Chicago's two-goal margin.
Los Angeles battled back to take the lead with three straight goals by Marian Gaborik, Dustin Brown and Tanner Pearson. Gaborik's goal was his NHL play-offs-leading 11th.
But Chicago's Ben Smith evened the match early in the third period to send the game to sudden-death overtime.
"One little play can be the difference," said Blackhawks goalie Corey Crawford, who has yielded four or more goals in four straight games.
Crawford, who entered the game with a dismal 0.870 save percentage against the Kings, made 40 saves and snapped a three-game losing streak. Kings netminder Jonathan Quick stopped 40 of Chicago's 45 shots.
The Kings will have another opportunity to clinch the series in Game Six on Friday on home ice, where they dominated the Blackhawks in Games Three and Four.
"It's win-or-go-home for those guys - it's desperation time, we tried to match that as best we could and we fell short," said Kings forward Justin Williams.
"We have a great opportunity coming home. It's not easy; attaining your ultimate goal is never easy."
The series winner will battle either the Montreal Canadiens or the New York Rangers, who lead that series 3-2, for the Stanley Cup.
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