Brandon Saad held the puck along the right wing below the circles and dished ahead to Handzus, who controlled the disc for a second before lifting a backhander under the crossbar to end the contest.
Saad ended up with a goal and two assists for the Blackhawks, who wasted a pair of early two-goal leads and fell behind by a goal before rallying to extend the series to a Game 6 in Los Angeles.
Corey Crawford looked shaky at times, but emerged with the win thanks to 40 saves.
Dustin Brown, Marian Gaborik, Jarret Stoll and Tanner Pearson lit the lamp for the Kings, who have the comfort of home ice on Friday to try and reach the Stanley Cup Finals for the second time in the last three seasons.
Jonathan Quick took the loss despite stopping 40 pucks.