Joe Thornton tied the game earlier in the third and got his second assist on the game-winner for the Sharks, who have won seven of eight to move two points ahead of Anaheim, although the Ducks have a game in hand.
The Sharks, who trailed by 13 points on Jan. 15, have not been alone in first since Dec. 6.
Patrick Marleau also scored and Antti Niemi made 26 saves for San Jose.
Teemu Selanne and Mathieu Perreault scored for the Ducks, who have lost six of their last eight. Frederik Andersen made 25 saves.
Burns, who assisted on Thornton’s goal, was on the receiving end on the game-winner when he crashed the net and deflected Thornton’s centering pass from the boards past Andersen.
Selanne gave the Ducks the lead early in the third shortly after Anaheim killed off a San Jose power play. He started the play with an aggressive forecheck and Ryan Getzlaf took the puck away from Marty Havlat along the boards. Corey Perry got the loose puck and his quick shot deflected off Selanne and into the net to make it 2-1.
But San Jose tied it midway through the period when Burns shot the puck off Andersen’s pads on a 2-on-1 and Thornton knocked in the rebound.
The game had a playoff-type atmosphere as the fans at the Shark Tank were loud and on edge from the start, breaking into derogatory chants about the Ducks right after the opening faceoff.
Neither team gave the opposition much space to operate in early in the game with only 10 combined shots in a tightly played first period that ended with the Sharks up 1-0.
The Sharks capitalized on their one power-play chance when Luca Sbisa was sent off for roughing Tommy Wingels. Sbisa got away with one questionable hit on Wingels before shoving Wingels to the ice a second time, leading to the penalty.
Thornton then sent a pass from behind the net to Marleau, who beat Andersen with a one-timer for his 29th goal. That was just the fifth power-play goal for San Jose in the past 21 games and the first point on the man advantage for Thornton in exactly two months.
Anaheim’s struggling power-play unit struck in the second, getting the team’s third man-advantage goal in the past 15 games. Selanne drew a pair of penalties from Logan Couture and Marc-Edouard Vlasic to give the Ducks a two-man advantage for 6 seconds.
Perreault won a faceoff from Thornton and then got in position to one-time a rebound of Corey Perry’s shot for his 15th goal just seconds after Couture left the penalty box.