Elias' second goal of the game was the 348th of his career, moving him past John MacLean for the top spot in Devils history and spoiling the debut of new Montreal head coach Randy Cunneyworth.
Earlier on Saturday, the Canadiens fired Jacques Martin and named Cunneyworth as the interim coach. The team is now just 13-13-7 and sits in last place in the Northeast Division after dropping its second straight game.
Elias scored the game-winner 1:31 into the third. Zubrus caused a turnover in the Montreal end and Sykora and Elias worked a give-and-go before Elias slammed it home for his 13th of the year.
Skating on the power play, the Devils took a 5-3 lead when Zubrus busted in through the middle and beat Price with a high wrister at the 10:18 mark.
Brodeur stopped nine shots in the final frame to cap the win.
The first goal of the game and only one of the first period went to New Jersey as good work along the left boards saw Zubrus get the puck back to Sykora, who threw it on net and got it in with traffic in front at the 7:05 mark.
Montreal tied the game 2:21 into the second on the power play thanks to a one- timer from Subban at the top of the left circle.
Elias, though, tied MacLean's record 2 1/2 minutes later with a power-play marker as Ilya Kovalchuk, from the top of the left circle, saw Elias streaking in down the right side and snuck a pass over there for the easy redirect.
However, Montreal scored the next two goals to take the lead as a slap shot from Campoli at the right circle tied the game at the 6:33 mark; and Eller teed up a shot in the slot and blasted it past the glove of Brodeur for a 3-2 lead 8:57 into the second.
Clarkson, though, tied the game for New Jersey with 1:34 to play in the second as he was sprung with a lead pass and beat Price with a low shot from the right wing.
Montreal hits the road for six games starting in Boston on Monday...New Jersey hosts the Rangers on Tuesday...Montreal had taken a 2-1 win in New Jersey last Saturday in the only other meeting between these two this season...The Devils went 2-for-5 on the power play while Montreal was 1-for-4.