San Jose, which also won all three meetings during the 2013 regular season, has not lost to the Canucks (3-2-0) since Jan. 21, 2012.
"We're just deep, we're really, really deep and we want to come at you every night," said Sharks captain Joe Thornton, who along with Logan Couture, added two assists. "We have been doing that early on."
The key was Marleau's goal that made it 3-1 midway through the second period.
Despite trailing 2-0 after the first, the Canucks managed to cut the San Jose lead in half with Mike Santorelli's fourth goal in three games at 10:20, only to have Marleau score on a one-timer just 95 seconds later.
"No matter what happened or where we struggled, we crawl back into it (at) 2-1. We need to at least play with that for a little bit and not get scored on so quickly," Canucks coach John Tortorella said.
"Really that was a gimme. We were just too slow. We turn it over when we don't need to and then we don't react defensively after the turnover."
Canucks goaltender Roberto Luongo held his team in it with 25 stops, but said Vancouver needed a better start.
"Obviously the first period hurt us. We've made comebacks the last couple of games, but against good teams like that it's tough to get out of a hole," he said. "I thought our first period we were a little bit tentative.
"Against a team like that you got to play 60 minutes if you want to give yourself a chance."
Sharks forward Tomas Hertl, who became the talk of the NHL after scoring four times in Tuesday's 9-2 win over the New York Rangers, was held pointless but was still a part of a dominant San Jose first line that includes Thornton and Burns.
San Jose led 3-1 after 40 minutes and had a good chance to extend the lead five minutes into the third, but Luongo kept Vancouver in it with a breakaway stop on Hertl.
The Canucks, who had come back from two-goal deficits against the Calgary Flames and the New Jersey Devils, showed little life until midway through the period.
Their new puck-pressure style eventually started to gain a bit of time in the Sharks zone, but they rarely threatened Niemi's crease.
"You've got to give them some credit. They've got a good team over there, play a great system," Canucks captain Henrik Sedin said. "You've going to have games like this where you have to win the one-on-one battles and we didn't do that enough."
Vancouver pulled Luongo with 3:08 to go in the period, but Burns scored into an empty net with 2:19 to play.
"They locked it down pretty well," said Tortorella, whose team had just six shots in the third. "It's not like we're playing against no one out there, but we need to have more people going to play against a balanced team like that."
Vlasic opened the scoring at 12:16 of the first period by finishing off a nice passing play. The San Jose defenseman took a pass from Burns off the rush in the high slot and snapped a shot blocker side on Luongo.