Toronto entered the game a point behind Columbus for the Eastern Conference's second wild-card spot and the gap remained at the end of the night after the Blue Jackets shut out the Philadelphia Flyers 2-0.
The Blue Jackets, however, have two games in hand on Toronto and the Maple Leafs have only four left to play.
"I know it is a cliche but we have our backs against the wall," said the Leafs' James van Riemsdyk, who notched his 30th goal of the campaign in the win.
"We have no choice, we just have to keep on clawing and scratching and find away to stay alive. Things are tough and we are going to have to take it one game at a time, one shift at a time."
After ending a seven-season playoff drought last year, the Maple Leafs, the only NHL franchise valued at a $1 billion by Forbes, look likely to once again be watching the postseason action on television.
The win was the second straight for the Leafs but it may be a case of 'too little too late' coming on the heels of an eight-game losing streak that sent them spiraling down the standings.
"A lot of pressure," said Kadri, knowing the Leafs needed a win against the NHL's top team to keep their playoff bid alive.
"We've never stopped believing, we have a couple of wins in a row and we're far from done."
Boston entered the game with a nine-point lead over Pittsburgh in the race for the number one seed in the East, and could have sealed the conference with a win over the Leafs combined with a Penguins loss to the Winnipeg Jets.
But Pittsburgh beat Winnipeg 4-2, trimming the deficit to eight after the Bruins picked up a point in the overtime loss.
"We don't want any easy games, we don't need that right now," said Bruins coach Claude Julien, eager that his team stays sharp through down the stretch.
"That's what we want to do from here on in."
Toronto opened the scoring when Paul Ranger was first onto a loose puck in front of the Boston net and swept it past goaltender Chad Johnson, but the Bruins hit back 56 seconds later on a goal from Brad Marchand.
With 12 seconds left in the first, Tyler Bozak knocked in Phil Kessel's centering pass and then 22 seconds into the second van Riemsdyk increased the Toronto advantage to 3-1.
Milan Lucic began the Bruins comeback by blasting his 22nd of the season past Jonathan Bernier five minutes into the third, while Patrice Bergeron forced overtime by beating James Reimer, who had taken over in the Toronto net after Bernier had to be helped off the ice with an apparent groin injury.
With the Leafs on a power play, Kadri kept Toronto's faint playoff hopes alive when he banged home a rebound midway through the extra session.
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