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San Jose, desperate to avoid going 2-0 down in the best-of-seven series, looked sharper from the outset with Tommy Wingels putting the Sharks ahead after just two minutes.
Wingels shot from the slot flew into the net after hitting the shoulder of goaltender Brian Elliott.
San Jose preserved the lead into the second period, only the second time in 16 playoff games that the Blues had trailed after 20 minutes.
Burns put the Sharks 2-0 up with his opening goal after seven minutes of the second period, converting a perfect setup from Joe Pavelski.
"Guys came out with a good sense of urgency and got after it," said Pavelski. "It was a great job by everybody."
Burns doubled his personal tally early in the third period while Troy Brouwer was serving a high-sticking penalty.
Dainius Zubrus added an empty-net goal with 19 seconds left in the game.
Martin Jones, who thwarted the Nashville Predators in Game 7 of the conference semifinals, had an easier time of it on Tuesday.
The Sharks goaltender was required to stop only 15 shots in the first two periods.
Jones finally finished with 26 saves, while Elliott stopped 20 shots.
"We wanted to go home with a split and that's what we got," said Sharks defenseman Marc-Edouard Vlasic.
Sharks coach Peter DeBoer said his team's ability to turn power plays into goals had been decisive.
"I thought we brought our good game to the rink tonight for longer periods," said DeBoer.
"Getting the first goal was big, and I knew our power play would bounce back. It always has all year."
Blues coach Ken Hitchcock was at a loss to explain the defeat.
"We didn't play very well," Hitchcock said. "I'm not sure why. I don't know if the players know why. They (the Sharks) were much better than us in every aspect, especially special teams. We tried to play the way we did in Game One and we didn't get away with it."
The series heads to San Jose for Game Three on Thursday.