Koivu scored past Tampa Bay's All-Star forward on a goal-mouth scramble 3:15 into the extra period and the Montreal Canadiens recovered from blowing a two-goal lead in the third for a 3-2 win over the Lightning on Thursday night.
Koivu put a rebound in through traffic after St. Louis stuck out his left palm in an attempt to stop Koivu's first shot when he wound up in the right side of the crease beside Ramo.
"When I got the rebound right after, I knew that the goalie was down and that he couldn't recover and I was hoping to hit something to go in, and lucky enough it went in," Koivu said.
Like Koivu, St. Louis believed the Lightning goalie made the initial stop.
"I was on my knees and I couldn't poke check the puck or anything," St. Louis said of his impromptu goaltending effort.
The Montreal captain danced off to the corner where he was joined by his teammates, who celebrated after blowing a 2-0 lead in the third.
"It was just a big point for us and at this point you've got to have the emotions to get it done, and when it happens, when a team wins in overtime, it's pretty big," Koivu said.
The Canadiens won their second in a row following a five-game losing streak to strengthen their hold on the eighth and final playoff spot in the Eastern Conference. They have 85 points.
Lukas Krajicek drew the Lightning even at 2 with his second goal of the season with 5:17 left in regulation.
The Canadiens outshot Tampa Bay 36-19 overall, including all three shots in overtime.
"When we're chasing pucks, we kill ourselves," Ramo said.
"Preferably, we would have kept the lead," Canadiens coach Bob Gainey said. "We created the lead and would have kept it, but I liked the way we played the game and we need the points -- I suppose that is the most important, we need the points -- but if not equally, almost as important is that we played a much more stable, much more consistent and reliable minute by minute throughout the game."
Kovalev reached the 20-goal plateau for the 12th time in his 17 seasons in the NHL when he scored on a power play at 11:19.
Latendresse brought the crowd to its feet when he flattened Evgeny Artyukhin with an open-ice check early in the third, and again moments later with his 12th goal on a wraparound play at 5:48.
The big left wing drove down the left side and carried the puck behind the net to reach around and tuck it into an open right side for his 12th goal.
Price, who faced only two shots in the first, made 17 saves for his second straight win.
The Canadiens are 3-3-2 in eight games since Gainey took over behind the bench after firing head coach Guy Carbonneau on March 9.
David Koci scored his first NHL goal midway through the third for Tampa Bay, which has lost four of five.
Kovalev opened the scoring. He took Alex Tanguay's cross-ice pass and beat Ramo with a shot from the left side.
Tanguay extended his points streak to four, including the third five-point game of his career in Tuesday night's 6-3 win over Atlanta. He has three goals and five assists during his streak.
Montreal outshot Tampa Bay 11-2 in the first and 13-6 in the second.
Schneider, who also assisted on Kovalev's goal, hit the right post with a slap shot from the point on a power play late in the second.
Tanguay missed 28 games through the first two months of the season after he injured his shoulder when he was driven into the boards by Artyukhin during a 2-1 shootout win in Tampa Bay on Dec. 30.
The Lightning fell to 5-17 in non-regulation games this season. ... Tampa Bay's two shots in the first were the fewest allowed by Montreal in a period this season. The Lightning were held to a season-low one shot in the first period of a 4-1 loss in Carolina on Feb. 20. ... Canadiens right winger Matt D'Agostini played in place of Andrei Kostitsyn, who had the flu. Right winger Georges Laraque, left winger Gregory Stewart and defenseman Ryan O'Byrne also did not dress for Montreal.