Tuukka Rask made 29 saves for the Bruins, who finished the regular season with the best record in the NHL but lost home-ice advantage in the series.
The Bruins, who never led in the game, outshot the Canadiens 14-6 in the third period and twice rallied to tie it. The Canadiens picked up a questionable power play at the end of the first overtime and failed to score, but there was little doubt about the call when Matt Bartkowski was sent off for taking down Dale Weise in the crease to keep him from getting to a loose puck.
Montreal won the ensuing faceoff and worked the puck around to Subban, who slapped it past Rask to end the game.
Game 2 is Saturday in Boston.
The 34th postseason series between the two Original Six rivals began without much of the intensity of past matchups. Only after the Bruins fell behind 2-0 did they seem to realize that they were in trouble.
When Josh Gorges was sent off for holding 14 minutes into the second period Boston's first power play of the game the Bruins repeatedly tested Price, holding the possession in the Canadiens' zone virtually the whole 2 minutes and getting several good scoring chances.
Dougie Hamilton hit the crossbar and Jarome Iginla missed a wide-open net, leaving Montreal's 2-0 lead intact heading into the third period. But no longer were the Bruins being pushed around.
Reilly Smith wristed one past a screened Price with less than 3 minutes gone in the third period to make it a one-goal game. Shortly after another unsuccessful power play, Milan Lucic muscled his way through the defense and passed it to Torey Krug for a slap shot that went past Price's left skate.
O Second round
• Montreal 4, Boston 3 (2OT)
Canadiens lead series 1-0
• N.Y. Rangers at Pittsburgh, 5 p.m., NBCSN
• Minnesota at Chicago, 7:30 p.m., NBCSN