The Maple Leafs are still five points behind eighth-place Buffalo and two behind the Carolina Hurricanes and will likely need to win all five of their remaining games to have any chance.
"We've just got a bunch of young guys in here right now ... and guys aren't really remembering what [the Leafs] did two or three years ago," said forward Darryl Boyce. "We're just worrying about the present and we're just going to keep rolling along."
A loss to the Sabres would have virtually ended their hopes on Tuesday. Toronto has beaten Buffalo three times since the All-Star break.
Before that, Buffalo was 29-9-1 in the series.
"Often you beat yourselves before the game's even started because you don't know if you can beat their goalie or beat their team," rookie goaltender James Reimer said. "When you do get that win, you kind of realize: 'Oh, these guys are beatable.'
"It shouldn't be like that, but I think sometimes it is," he said.
It was a tense final period at Air Canada Centre with the Leafs clinging to a one-goal lead. Reimer took one shot off the mask during a power play and managed to fall on the puck with plenty of traffic in front.
Phaneuf got Toronto off to the kind of start it needed, skating into the Sabres end and beating Ryan Miller with a hard shot at 1:34. The Sabres goaltender was left staring in disbelief at his glove.
"I had a chance to set the tone on my end and didn't do it," Miller said. "It just kind of went from there. ... [The shot] was rising up and I didn't have a step on it. So I just look stupid."
The Leafs extended the lead to 2-0 at 10:46 after Kadri stripped the puck off Shaone Morrisonn in the corner and worked out front to Boyce, who scored his fifth of the season.
It was one of the finest games of Kadri's young NHL career as he also set up the team's third goal after Vanek had put Buffalo on the scoreboard. The rookie skated around Tim Connolly and fed MacArthur to make it 3-1 at 5:06 of the second period.
The Sabres were quick to respond.
Pominville squeezed a shot past Reimer at 6:25 and Niedermayer stripped Carl Gunnarsson of the puck to tie it 3-3 just 30 seconds later.
Leafs coach Ron Wilson called a rare timeout to settle his team down and it seemed to work. Grabovski put Toronto back ahead at 13:35 of the second period by racing to the net and directing a Phil Kessel pass behind Miller for his 29th goal of the season.
"We're going to try and win all of our games and see what happens," Wilson said. "If we don't get in, we don't get in. But we'll have given it our best shot."
Fredrik Sjostrom and Brett Lebda were healthy scratches for Toronto. ... Phaneuf has four goals in his last eight games. ... It was the 14th time this season Buffalo scored two goals less than 60 seconds apart. ... MacArthur hosted 40 athletes from Special Olympics in a private box.