Dustin Penner and Drew Doughty also lit the lamp for the Kings, who have won the first three games of a four-game road trip. Jonathan Quick made 21 saves and was bailed out by his teammates after giving up the tying score with 43 seconds left in regulation.
Brad Richards and Jamie Langenbrunner scored for the Stars, who were 6-0-2 in their previous eight games. Jamie Benn had scored a goal in six straight, but failed to find the back of the net on Sunday.
Kari Lehtonen turned aside 27-of-30 shots in defeat.
Dallas entered the third period down by a goal, 2-1, but failed to panic due to its success with comebacks this season. The Stars came into the game with a 9-18-3 record when trailing after two frames.
It seemed all but over with the clock showing less than a minute. But a shot from the left point was not fully controlled by Quick. Langenbrunner spotted the loose puck under the pads of a sprawling Quick and chipped the disc into the net with 42.1 seconds left.
Los Angeles wanted no part of overtime, however, as Handzus found the back of the net with 20.8 ticks remaining. A battle for the puck at the side of the net was won by Alexei Ponikarovsky, who backhanded a pass to the slot. Handzus was right there to jam the puck past Lehtonen for the winning score.
The Kings drew first blood on the power play at the 3:04 mark of the first period. Doughty’s slapper from the right point beat traffic in front and got past Lehtonen.
Penner made it a 2-0 game at the 14:01 mark of the middle stanza. Anze Kopitar controlled the puck at the right circle and slid a pass in front to an open Penner, who chipped the disc into the net.
Dallas cut its deficit to 2-1 with two minutes remaining in the second. Loui Eriksson fed a perfect pass to a charging Richards, who beat Quick on a breakaway.
The Kings have taken three of five from the Stars this season…Dallas was without forward Adam Burish (lower body)…Los Angeles went 1-for-3 on the power play, while the Stars went 0-for-3 on the man advantage.