All the scoring took place in the middle period, the last two goals coming from Montreal on an extended power play.
Erik Cole was called for a major boarding penalty in the second period and during the 5-minute stretch while he was off the ice, both Andrei Kostitsyn and Picard scored to turn a 2-1 deficit into a 3-2 lead.
Montreal had lost five of its previous six games, including the first two of a season-long, seven-game road trip.
Carey Price got the win with 22 saves. Cam Ward stopped 26 shots for Carolina, which has lost two in a row after winning four straight.