The Kings relied on gritty defense and solid goaltending to silence the Hawks, and notched four goals in just over 10 minutes, wiping out what had been a 2-0 deficit.
The seven-game series now shifts to California for Game Three on Saturday.
"Our goal coming in here was to at least steal one. It's tough to win in this building," said Kings defenseman Drew Doughty, who assisted on the game-tying goal in the opening minutes of the final frame.
The Blackhawks opened the scoring with a high backhand goal by speedy defenseman Nick Leddy in the first period and extended their lead with a tally by winger Ben Smith early in the second.
But Los Angeles clawed back into the game after goalie Jonathan Quick went post-to-post to deny a slapshot by Blackhawks defenseman Brent Seabrook on a two-on-one break and the momentum shifted.
Kings' forward Justin Williams halved Chicago's lead late in the second period before the visitors' third-period onslaught saw Jake Muzzin, Tyler Toffoli and Jeff Carter all score to seal the victory.
Carter completed a hat-trick on an empty-net goal after the Hawks pulled goalie Corey Crawford with nearly four minutes left in the game.
"That save and scoring that goal late in the second period were big because it gave us some momentum going into the third. Obviously, the two power play goals don't hurt at all," said Kings forward Anze Kopitar.
Quick, who earned the Conn Smythe Trophy as the most valuable player of the play-offs during the Kings' 2012 championship run, stopped 23 of 25 Chicago shots.
Crawford made 25 saves but the six goals was the most allowed by Chicago during the play-offs.
"I don't think we've seen a game like that all year when we were doing everything right and then all of a sudden it was a disaster," said Blackhawks head coach Joel Quenneville.
Chicago are seeking a third Stanley Cup in five years, while Los Angeles are trying for a second in three years. The winner of the series face either the New York Rangers or Montreal Canadiens.
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