Price fended off Carolina early before Montreal rallied for a 4-3 shootout win on Wednesday night.
Brian Gionta scored the lone goal in the shootout and Price stopped all three Carolina shooters as Montreal picked up its third win in five games.
The Canadiens improved to 10-9-3 and snapped Carolina's two-game winning streak.
The Canadiens also beat the Hurricanes 4-0 on Nov. 16 at the Bell Centre
"They were definitely thinking about the last time we played them," said Price, who finished with 31 saves.
"They were probably looking for a little bit of retribution and we knew they would come out hard and play well."
Tlusty's goal came 34 seconds into the game after his shot near the blue line glanced off the left post and skidded along the goal line behind Price. The puck almost went all the way through, but Price fell backward and knocked the puck into his own net.
"It rolled right along the goal line and I saw their guy charging the net so I tried to lay out and I knocked it in with my stick," Price said.
When Stewart made a one-timer from the slot to give Carolina a two-goal cushion at the 6-minute mark, the Hurricanes seemed to have all the momentum.
Montreal fought back to tie the game by getting a short-handed goal from Lars Eller at 9:06 of the second and then the Canadiens knotted the score at 2-2 when Michael Cammalleri tipped in a pass from Josh Gorges 27 seconds into the third period.
"In the first period we weren't good," Montreal coach Jacques Martin said. "They won more battles and were on the puck. They were more determined. After that we came back strong in the second, created several good chances, capitalized and persisted as we should have."
In overtime, Carolina captain Eric Staal had a solid shot bounce off the post with 1:30 left. The Canadiens also had a pair of chances in the extra period that Carolina's Cam Ward, who had 30 saves, stopped with stellar goaltending.
Montreal D Hal Gill returned to the lineup after missing the past five games with an infection. ... With cousins Brandon and Brett Sutter picking up assists for Carolina, it marked the first time since March 3, 1994, that two Sutter teammates registered points when Brent had two and Rich got one for Chicago against Calgary. ... Montreal got its second short-handed goal of the season on Eller's tally in the second period.