Matt D'Agostini, Kostitsyn and Begin scored in the opening period for the Canadiens, who ended a three-game losing streak. Guillaume Latendresse also scored for the second game in a row late in the third.
Jaroslav Halak made his fifth straight start for Montreal and stopped 30 shots.
Arron Asham scored in the first and assisted on Braydon Coburn's goal in the third for Philadelphia, which had not lost in regulation in seven games (6-0-1).
Canadiens defenseman Mike Komisarek returned to the lineup and assisted on Kovalev's goal after missing 16 games because of a shoulder injury.
D'Agostini opened the scoring with his fifth goal 6 minutes in.
The rookie right wing got behind Flyers defenseman Matt Carle to receive Maxim Lapierre's long cross-ice pass from the Canadiens' zone and drove the left side before cutting toward the net to beat Antero Niittymaki with a quick shot for his fifth goal in nine games.
Kostitsyn made it a two-goal lead with a power-play goal at 10:33 when he took Robert Lang's return pass and one-timed a slap shot past Niittymaki from the left point.
Kostitsyn got his second straight point with an assist on a goal by Begin, who increased Montreal's lead to 3-0 when he deflected Tom Kostopoulos' shot from the right side for his fifth of the season at 15:26.
Asham scored 32 seconds later to draw Philadelphia to 3-1. D'Agostini hit the right post when he shot off a turnover in the Flyers' zone early in the second.
Kovalev restored the Canadiens' three-goal margin at 6:20 when he drove in on a breakaway and put a low shot past Niittymaki inside the right post.
Coburn scored his third goal 13:28 into the third to draw Philadelphia to 4-2.
Latendresse scored for the second game in a row on a power play at 18:57.
Montreal played the first of three straight home games. The Canadiens are on a 5-2-1 run at the Bell Centre during a stretch of 10 of 11 home games that includes a franchise-record seven-game homestand that ended Saturday.
Notes: Asham's point shot broke a pane of glass above the boards directly behind the Canadiens' goal 15:14 into the third. The game was delayed for 6 minutes while the maintenance crew replaced it and cleaned up the ice. ... Philadelphia C Mike Richards ended an eight-game points streak. The Flyers captain had 15 points during that stretch, including seven goals and eight assists. ... Kovalev has 325 goals and 475 assists in 1,104 regular-season games with the New York Rangers, Pittsburgh and Montreal. ... Canadiens LW Andrei Kostitsyn missed a couple of shifts in the second period and did not come out for the third.