Brian Boucher, taking over No. 1 goalie duties from injured Ray Emery, made 22 saves for the Flyers, who broke a five-game losing streak. Philadelphia has won 12 in a row and 15 of 16 against the Islanders, dating to the 2007-08 season.
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.
Islanders rookie John Tavares scored both goals for New York, which dropped its second in a row and slipped into a last-place tie with the Flyers in the Atlantic Division.
Martin Biron stopped 31 shots for New York.
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.
CANADIENS 4, SENATORS 1
In Ottawa, Jaroslav Halak made 45 saves, and Tomas Plekanec had three assists as Montreal extended its winning streak to three with a win over Ottawa.
Mike Cammalleri, Marc-Andre Bergeron and Andrei Kostitsyn all scored for the second night in a row, and Scott Gomez added an empty-net goal.
Cammalleri scored for a third consecutive game late in the first for Montreal, which beat Philadelphia 3-1 on Monday despite being held to 13 shots.
Mike Fisher scored his 13th goal early in the second for Ottawa, which had a five-game, home-winning streak snapped despite a 46-27 shots advantage.
PREDATORS 4, CANCUCKS 2
In Nashville, Tenn. Martin Erat had his second career NHL hat trick and added an assist to lead Nashville to a win over Vancouver.
The Predators beat the Canucks at home for the first time since Nov. 23, 2006.
After winning the first two games of a four-game trip, the Canucks lost at Carolina and Nashville.
Patric Hornqvist had the other goals for the Predators. Ryan Kesler and Daniel Sedin scored for Vancouver.