Brian Boucher, taking over No. 1 goalie duties from injured Ray Emery, made 22 saves for the Flyers. Philadelphia has won 12 in a row and 15 of 16 against the Islanders, dating to the 2007-08 season.
"It feels nice to win a hockey game, and finally get a couple bounces," captain Mike Richards said. "We executed a lot of plays. We were finishing chances."
The Flyers had dropped two consecutive games - including an 8-2 home loss to Washington on Saturday - after Laviolette replaced fired coach John Stevens on Friday. His first win with his new team coach came against the club that gave him his first head-coaching job, from 2001-03.
Islanders rookie John Tavares scored both goals for New York, which dropped its second straight and slipped into a last-place tie with the Flyers in the Atlantic Division.
Richards, scoreless in the previous four games, netted his first goal of the night on a power play, snapping a five-game, 0 for 19 stretch with the man advantage for the Flyers. Richards deflected Kimmo Timonen's slap shot past goalie Martin Biron, who spent the previous three seasons with Philadelphia.
Just 6 minutes later, Richards scored the first short-handed goal allowed by the Islanders this season. He stripped the puck from defenseman Mark Streit and flicked it past Biron's stick side during a breakaway.
Carter netted the next two goals to push the Flyers' lead to 4-0. Danny Briere fed him a pass on the left goal line, and Carter angled a wrist shot past Biron. His second goal came on a rebound in front of a crowded net.
Tavares tried to rally the Islanders with a pair of power-play goals that cut New York's deficit in half. His first was scored at 10:52. He made it 4-2 with his 13th this season, scored at 14:55.
Philadelphia nearly let New York back in the game by taking five penalties in the second period. The Flyers pushed their lead to 5-2 on Giroux's first of the night with 2:31 left in the frame.
The Flyers (14-13-1) were 12-5 on Nov. 18, but their slump led to Stevens' dismissal. Philadelphia held a 37-24 shots advantage, one night after they had a 15-13 edge at Montreal in a 3-1 loss to the Canadiens.
"It certainly was good to get some opportunities tonight and have some guys score some goals where they haven't been," Laviolette said. "There haven't been a lot of them lately. I thought that a lot of our game had that in it last night, but we just didn't bring the rubber to the net. Tonight we did a much better job of that. I thought that our key players were pretty sharp."
The 4-0 lead helped Boucher relax. He had started five of the previous eight games, while Emery tried to recover from his injured abdomen. The 32-year-old Boucher will now be counted on to shoulder the load.
"I try to prepare the same way every day," he said. "In your mind, you're ready at all times. For me, it's just a matter of getting to play. Hopefully, I can provide the solid goaltending that's needed for us to win games.
"We had a five-game losing streak, and you have to stop the bleeding at some point."
The Islanders, meanwhile, continue to find ways to lose to Philadelphia.
"We don't read too much into that stuff," coach Scott Gordon said. "Every game is different: different players, different goalies."