The Leafs outshot Montreal 51-22, including 8-2 in overtime, but needed Kyle Wellwood's goal on the seventh shot in a shootout to pull out a 5-4 win on Saturday night.
The Canadiens (5-2-3) got one point because Sergei Samsonov, with his second of the game, and Sheldon Souray scored in the final 6:16 of regulation time to force overtime and the shootout.
"We were extremely lucky," said Carbonneau. "I'm happy we got a point, but I'm not very happy with the performance from start to finish."
Leafs coach Paul Maurice agreed.
"I thought we were the best team for the entire game and I would have felt ripped off if we didn't pull it out," he said.
The star of the game was defenceman Tomas Kaberle, the target of critics after 6-2 and 7-2 losses to Ottawa this week, who had three goals and an assist in regulation time and scored on Toronto's first shot of the shootout.
It was the first hat-trick by a Toronto defenceman since Borje Salming in October, 1981.
"He had a lot of heat and pressure coming into the game and I liked the fact that he came out and played his best game of the season," said Maurice.
"You guys (the media) were on us big-time, so it's nice to go on the road and get away from it," added Kaberle.
Kaberle beat David Aebischer on the opening shot of the shootout, but Samsonov replied with a goal.
Five players from each team then were stopped before Wellwood beat David Aebischer with a deke to the forehand and Jean-Sebasien Aubin stopped Montreal's Alexander Perezhogin.
Aebischer had a shaky start, but then came up huge for the Canadiens as they were outskated by the Leafs (5-4-3) and couldn't get the puck out of their zone for long stretches.
Carbonneau was stunned by how flat his team played against Montreal's oldest rival before a full house of 21,273 roaring, chanting fans at Bell Center.
"We weren't playing Columbus, it was the Toronto Maple Leafs," said Carbonneau. "They beat us to every loose puck."
"They decided to play a game based on speed and they looked like (first-place) Buffalo and we weren't moving."
But Samsonov's second goal of the game and a goal on a rocket shot from Souray tied the score with 3:44 left in regulation time to earn a point. Perezhogin, with his first of the season, also scored for Montreal.
Samsonov had been much in the news all week when he reacted with anger to being demoted to checking lines in the previous two games.
But Carbonneau made a new line with Alexei Kovalev at centre between Samsonov and Perezhogin and it worked. Kovalev had three assists.
"I don't think I have to prove anything," said Samsonov, the team's main free-agent signing this summer. "I feel that every game I play I give 100 per cent and whatever happens, happens."
"This was just one of those games where I had an opportnity to score and I did. But I'm not trying to prove anything."
The Leafs' 4-2 lead in the third period looked safe until a shot off the end boards went to Samsonov's stick. He tucked the puck in behind Aubin at 13:44 to cut the Leafs lead to one.
With the crowd in a steady roar, Mark Streit slipped the puck to Souray on the left point to move in and blast his sixth goal of the season past Aubin to tie the game and force overtime.
Early on, it was all Toronto.
Kaberle, who had only two assists in the Leafs' first 11 games, assisted Ponikarovsky's power play goal only 1:44 into the game after Saku Koivu slashed Michael Peca 30 seconds into the match.
A turnover in the Montreal end allowed Kaberle to fire a wrist shot past Aebischer at 4:50.
Samsonov redirected a Streit pass behind Aubin at 7:39.
But Kaberle's backhand dribbled through Aebischer after some dogged work near the net by Darcy Tucker at 19:17.
A rare Montreal transition saw the speedy Perezhogin go around defenceman Ian White on a rush and slip the puck in the far corner at 10:39 of the second frame.
Kaberle got that one back, too, at 12:05 when he threw the puck into a crowd in front and saw it sneak under Aebischer. The goal stood up after video review showed it was in.
It was Aubin's second start this season. He started in a 7-6 shootout loss in New Jersey on Oct. 12.
The win evened the series between the rivals this season. Montreal defeated the Leafs in a shootout at Air Canada Centre on Oct. 7. The teams split their eight meetings last season.
The Leafs, now 2-3 this season and 5-10 all-time in shootouts, return home to face Atlanta on Monday night and then head off on the road to face Tampa Bay, Florida and Buffalo.
The Canadiens stay home to play Ottawa on Tuesday.
Notes: - Nikolai Antropov returned to the Toronto lineup after missing the first 11 games with an ankle injury. . . Forwards Wade Belak, Ben Ondrus and Alexander Suglobov were healthy scratches for the Leafs. . . Montreal is missing D Mathieu Dandenault (leg), D Francis Bouillon (knee) and F Aaron Downey (concussion). . . The Canadiens are 319-266-88 all-time against Toronto.