The defending Stanley Cup champions continued to rebound from a slow start in which they lost seven of their first 10 games.
The Bruins let a two-goal lead slip away in the opening period, but Marchand put Boston ahead to stay with a power-play goal 3:50 into the second.
Marchand produced the game's final goal with 2:03 left in the third, 44 seconds after Milan Lucic had scored on the power play to give the Bruins a 5-3 lead.
Johnny Boychuk and Jordan Caron tallied 1:44 apart in the first period to put the Bruins in front, but Edmonton came back to tie it before the period was out on goals by Ryan Smyth and Ryan Nugent-Hopkins.
Tuukka Rask made 27 saves to get the win. It was his second victory in five appearances this season.
Devan Dubnyk gave up all six Boston goals on 28 shots.