Chicago handed the Kings their first home playoff loss this year and took a 3-1 lead in the championship series of the Western Conference. The Blackhawks, who had the best regular-season record in the NHL, will have a chance to earn a spot in the Stanley Cup Finals when they host Game 5 on Saturday.
Defending champion Los Angeles had won its first eight home games in the postseason, including a 3-1 decision over the Blackhawks on Tuesday.
The Kings twice took the lead in Thursday's game, coming up with goals from Slava Voynov and Dustin Penner early in each of the first two periods.
Hossa's game-winner was his seventh goal of the playoffs and Corey Crawford blanked the Kings the rest of the way. He had to make just 19 saves to get the victory, improving his playoff record this season to 11-5.
Jonathan Quick stopped 25-of-28 shots for the Kings.
The first spot in the Stanley Cup Finals could be decided on Friday when Boston hosts Pittsburgh in Game 4 of the Eastern Conference title series. Boston will take a 3-0 lead into that game.