Wesley's goal with 5:22 left in the second period lifted the Hurricanes into the deadlock. The Canadiens got 29 saves from Jeff Hackett but continued to struggle on the road.
Wesley put Carolina in a 1-0 hole 6:50 into the second period when enforcer Gino Odjick's centering attempt from the right side deflected off the defenseman's stick and got past goaltender Arturs Irbe.
It was Odjick's first goal since Jan. 6, 2001, a span of 14 games.
Wesley atoned for that gaffe just under eight minutes later. With teammates Jeff O'Neill and Ron Francis screening Hackett, he scored his fourth goal of the season on a wrist shot from the high slot.
Carolina had all four shots in overtime, but Hackett stopped them all in just his second start in nine games.
Irbe made 19 saves for the Hurricanes, who played for the first time since trading All-Star defenseman Sandis Ozolinsh to the Florida Panthers in a five-player swap. Defenseman Bret Hedican and center Kevyn Adams, two of the three players who came over from Florida, made their debuts for Carolina.
The Hurricanes are 1
Montreal is 1-6-2-1 in its last 10 on the road.