Chris Minard and Tyler Kennedy also scored a goal each for Pittsburgh, which saw captain Sidney Crosby return to the ice. He did not play in Friday's win over Anaheim after suffering a lower body injury Wednesday against Washington.
Fleury's shutout is his second this season for the Penguins, who have won two in a row and three of four.
"We got some big goals and Fleury played great," said Crosby. "We didn't give up a ton of scoring chances, but when we did he was there and shut the door."
Henrik Lundqvist gave up three goals on 23 shots for New York, whose three- game win streak came to an end. They've lost in six of their last seven trips to Pittsburgh.
"Home ice is an advantage," said New York forward Chris Drury. "We have a good record against them at home, but obviously not so good here. We obviously have to try and figure that out."
Pittsburgh had an early power play, but couldn't capitalize on several chances, particularly Jordan Staal. Toward the end of the man advantage, he took a drop pass and fired a shot off the left post, then was stopped on a breakaway backhander shortly after the power play had ended.
But Minard put Pittsburgh on the scoreboard at the 7:35 mark of the opening period. Sykora fired a backhanded pass from behind the net out front, and Minard one-timed it past Lundqvist for a 1-0 edge.
Then, with 4:28 remaining in the stanza, Kennedy's shot from the right circle beat Lundqvist stick-side for a 2-0 advantage.
Crosby had a penalty shot early in the second period, but was unsuccessful. However, Sykora's power-play marker at 7:32 made it 3-0 Penguins, and Fleury's goaltending made the lead stick.
He made 16 saves in the second period and 10 in the third to help Pittsburgh cruise to the win.