Kovalchuk scored New Jersey's winning goal for the second straight night. He did the same thing Thursday against Toronto late in overtime.
New Jersey won its fourth straight, getting 31 saves from Johan Hedberg.
Neither team could score over the first two periods, but Patrick Marleau gave San Jose the lead 6:08 into the third with his 21st goal of the season.
With San Jose's Logan Couture in the penalty box for tripping, Brian Rolston tied it midway through the final period. Kovalchuk then put a wrist shot past Antti Niemi for his 19th of the campaign.
Niemi stopped 21 shots to take the loss. San Jose had a season-best, five-game winning streak stopped. The Sharks have not won in New Jersey in 10 seasons.