The Maple Leafs increased their points total to 78, three behind idle Buffalo for the Eastern Conference's final playoff berth. Toronto has seven games remaining.
``We just have to get some wins ourselves and sooner or later we'll get some help,'' Toronto defenceman Luke Schenn said.
The Maple Leafs have been one of the NHL's hottest teams in the last seven weeks, going 15-6-5 since returning from the all-star break.
``It would have been nice to get a couple more wins at the beginning of the season to give ourselves a better chance right now,'' Maple Leafs goalie James Reimer said. ``We're happy with the way we're playing right now. I just hope we can keep it going.''
Colorado, meanwhile, saw its modest two-game winning streak snapped on a night when the Avalanche lost two more players to injury.
Defenceman Ryan Wilson did not return after suffering a second-period knee injury.
Colorado also lost Mark Olver, part of the team's top line, to a head injury midway through the first period. He was levelled by Mike Komisarek's hard check near the red line and lay sprawled for several moments before crawling off the ice.
Even before Olver managed to reach the Colorado bench, teammate Cameron Gaunce, recalled from Lake Erie of the American Hockey League earlier Thursday, was exchanging blows with Komisarek that resulted in 5-minute penalties for both for fighting.
Colorado coach Joe Sacco said while it appeared that Komisarek's hit was legal, the loss of Olver early in the game was another setback for the Avalanche in an injury-plagued season.
``Losing him doesn't help because he's been playing very well and he gives us a lot of energy,'' Sacco said. ``He'll be re-evaluated (Friday) morning, along with Ryan Wilson.''
The Maple Leafs went up 4-2 on a goal by Kessel that he drove past goalie Peter Budaj, who had replaced starter Brian Elliott at the beginning of the third period. Elliott, who had to have some emergency repairs on his skates, returned shortly after Kessel's goal at 3:02 of the third.
After officials waved off an apparent goal by Toronto's Mike Brown, ruling the puck had been kicked in, the Avalanche again closed to within a goal on a score by Matt Duchene. He beat Reimer with a rebound shot at 11:55 of the third. But in the final minute, with Elliott pulled in favour of an extra attacker, Reimer turned away a flurry of shots by the Avalanche.
``We found a way to claw back into it and we had a couple of chances when we pulled the goalie,'' Sacco said. ``But we weren't able to capitalize. I like the fact that we played to the end, but catch-up hockey is tough to play in this league.''
Toronto took a 3-1 lead when Tyler Bozak redirected Luke Schenn's shot into the open side of the net past Elliott at 9:37 of the second period. Colorado got back within a goal 3 minutes later when their power play delivered a score by Ryan O'Reilly.
Colorado opened the scoring on Daniel Winnik's short-handed goal 5:55 into the game, but Toronto went in front on a pair of goals by Kulemin.
Kulemin's first goal at 10:32 of the first went into the net off Elliott's pads. He also emerged with the puck following a scramble behind the net, squeezing off a shot that got past Elliott in the final seconds of the period.
Notes: Winnik's first-period goal was his 11th of the season, tying his career high set as a rookie while with Phoenix in 2007-08. ... The Maple Leafs and Avalanche met for the only time this season in what was the last of Colorado's 18 meetings against Eastern Conference opponents. ... Scoring his 27th and 28th goals of the season, Kulemin tied Mikhail Grabovski for second on the team, one behind the club leader, Kessel.
1 - 0 COL
1 - 1 Tie
2 - 1 TOR
3 - 1 TOR
3 - 2 TOR
4 - 2 TOR
4 - 3 TOR