The Maple Leafs are still five points behind eighth-place Buffalo, and two behind Carolina, and will likely need to win all five of their remaining games to have any chance.
A loss to the Sabres would have virtually ended their hopes on Tuesday.
Dion Phaneuf, Darryl Boyce and Clarke MacArthur had the other goals for Toronto. MacArthur also added two assists.
It was a tense final period at Air Canada Centre with the Leafs clinging to a one-goal lead. James 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.
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.
NOTES: 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.