Byron, who assisted on Galchenyuk's second-period goal, scored his third goal from the edge of the crease to give Montreal its third one-goal lead. Galchenyuk and Radulov assisted on the goal.
"You've just got to seize the opportunity and do what you can because it's not too often you get to play on the first line with those two guys, so you've got to make the most of it," Byron said.
Shea Weber and Galchenyuk scored 1:10 apart to give Montreal two one-goal leads in the second. Carey Price made 41 saves for the Canadiens (11-1-1), who lead the NHL with 23 points.
"We're getting timely goals," Price said. " ... It seems like we're just finding ways to squeak 'em out, and that's a really good sign."
WATCH: All Bruins vs. Canadiens highlights
David Pastrnak scored for a fifth straight game to draw Boston even at 2-2 on a power play at 6:58 of the third period.
"Those games are tough," Pastrnak said. "You play a pretty good game the whole 60 minutes and then you get one bad bounce and then you lose the game."
Colin Miller scored in the second period and Jake McIntyre made 20 saves in his second start for the Bruins (7-6-0).
"We played well as a team and so we obviously had a chance to win this," Boston coach Claude Julien said.
Price has won each of his eight starts this season. He extended his winning streak to 11, the longest in the NHL since New York Islanders goalie Jaroslav Halak, Price's former Canadiens teammate, won 11 consecutive games from Nov. 5 to Dec. 4, 2014, according to the Elias Sports Bureau.
Weber gave Montreal a 1-0 lead on the power play, scoring his fifth goal of the season at 3:58 of the second.
Miller tied it at 2-2 when he got credit for his first goal 50 seconds later. His shot went wide of the left post, but rebounded off the end boards and bounced into the net off the back of Price's right pad at 4:48.
Galchenyuk put the Canadiens ahead 2-1 when scored for a second straight game 20 seconds later when he drove to the net, took a pass from Radulov and beat McIntyre at 5:08.
