Christian Ehrhoff and Jeff Carter also scored and Jonathan Quick had 21 saves for the Kings, who won their fifth straight and improved to 6-0-1 in their last seven games.
Scott Hartnell and Matt Calvert scored for Columbus, which lost goalie Sergei Bobrovsky to an apparent injury with about 5 1/2 minutes remaining in the third period. Bobrovsky had 33 saves and Curtis McElhinney stopped five of seven shots he faced.
"I’ll tell you what impact Bob had on the game," Blue Jackets coach John Tortorella said. "We don’t have a chance. I mean we’re not even close if it isn’t for Bob. We’re down four or five goals going into that third period if we don’t have Bob."
There were no immediate reports on Bobrovksy’s condition and Tortorella said he will be evaluated on Wednesday.
Carter, the former Blue Jacket who was booed throughout the game, tied it with 1:27 left in regulation, sliding home his 11th of the season in a scramble near the crease with Quick pulled for an extra skater.
In overtime, the Kings played the patient possession game. On the winning combination, Dustin Brown passed to Nick Shore on the left side, and he sent it to Martinez, who had an open net.
"Brownie just kept holding on to that puck, cutting back tiring the other guys out and slipped it to down to Shorsey," Martinez said of the play leading to his fourth. "Shorsey made a good look at the net to get that D-man to bite a bit and slid it over to be. I just ended up being in the right spot."
Brown said it was about waiting for an opportunity. "I thought we did a really good job," he said.
The Blue Jackets continue to trade wins and losses while needing to string some wins together to move up the standings.
In the third period and down a goal pressing for the equalizer, the Kings ticked off the first eight shots before the Blue Jackets registered one with 6:27 elapsed.
"We just need to do a better job of capitalizing on our chances," Martinez said. "That said, if we weren’t getting them I’d be concerned."
Los Angeles carried the play but Columbus almost increased their lead. Quick closed the door with a stretching save on Ryan Johansen from point blank. Brandon Dubinsky rung the rebound off the right post.
After a scoreless first period, Los Angeles outshot Columbus 15-6 in the second period but found themselves down 2-0 before the Kings put one on the board late in the period.
Columbus quickly struck on its first power play. Nick Foligno, setting up shop near the net, sent a short pass to the low, left circle that Hartnell one-timed past Quick for his 11th at 2:32.
Calvert made it a two-goal lead on the Blue Jackets’ 11th shot. He took a long pass at the red line from Brandon Saad, and wristed a shot that bounced off Quick’s glove and in with 5:43 left.
Late in the period, the Kings had Columbus pinned in its own zone. Ehrhoff one-timed an errant clearing attempt and hammered a shot from the point by Bobrovsky.
"A good character win staying with it," Martinez said.
Notes:The Kings opened a five-game road trip after a perfect 4-0 homestand. … Kings LW Kyle Clifford was scratched late with an upper-body injury — potentially suffered Sunday in a 3-1 win over visiting Tampa Bay — forcing Los Angeles to play with one less forward. … Los Angeles plays nine of its next 10 games on the road. … Carter extended his points streak to seven games.