St. Louis fought back from a 4-0 deficit to trail by one heading into the third period, but a power-play goal from Burns with 6:41 to go in regulation ended the Blues' rally.
It was the Sharks' fourth win in a row. San Jose has lost just three times in regulation this season, the lowest such total in the NHL.
Burns scored on a slap shot 35 seconds into the game and tallied again midway through the first period. The Sharks' other scores in the opening frame came from Joe Thornton and Tommy Wingels.
Antti Niemi stopped 21-of-24 shots.
St. Louis got goals by Ian Cole, David Backes and Jaden Schwartz in their first loss after a five-game winning streak.