Mark Scheifele also scored and Connor Hellebuyck made 16 saves for the Jets, who outshot the Kings 39-17.
Kovalchuk got his first goal since returning from Russia with the Kings this season on a three-year deal. It was the 418th of his career.
Jack Campbell, replacing injured starter Jonathan Quick, stopped 37 shots in his ninth career game.
The 35-year-old Kovalchuk got behind Winnipeg's defensemen, skated to the net and redirected Drew Doughty's pass at 6:26 of the opening period for a 1-0 lead.
The Jets tied it four minutes later when forward Brandon Tanev spun away from a defender along the boards and sent a quick pass to the slot for Scheifele's first goal of the season at 10:59.
The Jets had almost back-to-back power plays near the midway point of the second period and were successful on the second one when Connor poked the puck in with his stick after Blake Wheeler's shot went off Campbell and began heading toward the goal line at 10:21.
Winnipeg's shot advantage in the second was 19-5.
Los Angeles had its first power play of the game midway through the third, but Hellebuyck made a big save and then got a piece of another shot that hit the crossbar.
Winnipeg defenseman Tyler Myers was called for high-sticking with 3:02 remaining. Campbell went to the bench with about 90 seconds left to make it 6-on-4 but couldn't convert with the advantage.
Kings: Play the second game of their four-game trip in Montreal on Thursday night.
Jets: Travel to Nashville for a game Thursday night.
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