Tomas Plekanec scored consecutive goals in the final three minutes, and Montreal regained the top spot in the Eastern Conference. The Canadiens lead the Pittsburgh Penguins by one point.
The Sabres squandered a 3-1 third-period lead for the second straight home game, and had their playoff hopes take another hit entering the final week of the season.
Stuck in 10th place, Buffalo is four points back of eighth-place Boston and the postseason cutoff in the East.
Higgins was the star in a game the Canadiens had no business being in after they managed 10 shots through 34 minutes.
With Montreal pressuring in the Buffalo end, Saku Koivu, from behind the net, directed an errant pass into the right circle to defenseman Mark Streit. Rather than shooting, Streit fed a perfect pass to Higgins who, standing in front of the left post, easily directed the puck behind sprawling goalie Ryan Miller.
Streit had a hand in Montreal's final three goals, and the Canadiens won their fourth straight.
The comeback began after the Sabres seemed to take control when Nathan Paetsch and Daniel Paille scored less than 2 minutes apart to put Buffalo up 3-1 a little over 7 minutes into the third period.
Plekanec made it 3-2 with a power-play goal, deflecting in Streit's shot from the blue line with 2:31 left. With Montreal's net empty, Plekanec tied it with less than 15 seconds remaining.
Streit's shot from the blue line hit defender Henrik Tallinder in front, and the puck dropped in the slot, where Plekanec reached back and swept it through Miller's legs.
The Sabres are running out of chances to make up ground. Buffalo's meltdown was reminiscent of its previous home game Tuesday, when it squandered a 3-1 third-period lead in a 6-3 loss to Ottawa.
