Caufield scored from along the goal line to make it 3-2 at 2:15 of the third period.
Foerster scored his first goal in 15 games to make it 4-2 at 10:09. The goal was upheld after the Canadiens challenged for goaltender interference.
Couturier scored on the power play to put the Flyers ahead 5-2 at 11:43. It ended Philadelphia's 0-for-34 drought on the man-advantage and was their first power-play goal in 14 March games.
Dvorak scored at 13:34 to pull Montreal within 5-3 but 18 seconds later Michkov made it 6-3 at 13:52 on a breakaway after a giveaway by Canadiens defenseman Mike Matheson.
The Flyers forward leads NHL rookies with 22 goals.
"That play in front of the bench, where he picks it off and goes down for the breakaway, that's an exceptional little cat and mouse game that he plays with that guy, and he wins it," Shaw said. "Huge goal. They get back within breathing distance, and he puts one five-hole. Really smart shot."
Laine closed the scoring at 16:47 for the 6-4 final.
"Maybe not the way that we draw it up," Couturier said. "We had some good looks, just can't finish. Maybe it'll kind of loosen us up a little bit for the upcoming opportunities we get.
NOTES: Flyers defenseman Cam York was on the bench but did not play because of a disciplinary issue. Shaw said he expected York to play his usual complement of minutes when the Flyers play the Buffalo Sabres on Saturday. … Shaw coached his first game since April 18, 2006, with the New York Islanders against the Flyers. He's the third head coach to go at least 18 years between regular-season games, along with 20-year spans for John Muckler (Jan. 19, 1969, with Minnesota North Stars to Oct. 5, 1989, with Edmonton Oilers) and Al MacNeil (April 3, 1982, to Dec. 3, 2002, with Calgary Flames). ... Couturier has eight points (four goals, four assists) during a season-high five-game point streak. ... Michkov has seven points (two goals, five assists) during a four-game point streak. ... Canadiens forward Nick Suzuki had an assist on Laine's goal to give him five points (two goals, three assists) during a five-game point streak. … Hutson leads NHL rookies with 56 points (five goals, 51 assists).