WATCH: All Avalanche vs. Maple Leafs highlights
Antoine Bibeau made 26 saves in his NHL debut for Toronto (11-11-5), which has lost four of its past five games.
"I was feeling really comfortable," Bibeau said. "I thought I would be more nervous. I think I was well-prepared for that game, I had a good week of practice. The boys made me feel comfortable in the room, and I felt really good once the game got going."
The Maple Leafs outshot the Avalanche 21-8 in the first period, but Varlamov allowed Colorado to take a 1-0 lead when Mikko Rantanen scored on the power play with 1:09 remaining. Rantanen took a pass from Matt Duchene and put it into an open net with Bibeau out of position.
The Maple Leafs had killed off 18 consecutive power plays before allowing the goal.
Varlamov was 0-4-0 in his four prior starts, allowing 14 goals in those games, and gave up six goals in relief of Calvin Pickard against the Canadiens on Saturday.
"Confidence, you know," Varlamov said when asked how he turned around his performance. "When you are a little bit struggling, it's hard sometimes to find that confidence. The team gave me confidence today when we scored the first goal."
Nathan MacKinnon scored his eighth goal at 7:27 of the third period to give the Avalanche a 2-0 lead.
Maple Leafs forward Leo Komarov was assessed a double minor for high sticking MacKinnon on the play, but Toronto killed off each power play.
With the Maple Leafs pressuring, Avalanche defenseman Patrick Wiercioch knocked the net off its moorings and was assessed a penalty for delay of game with 4:05 remaining. Colorado defenseman Nikita Zadorov was called for cross checking Nazem Kadri with 2:56 left, giving Toronto a 6-on-3 power play with Bibeau pulled, and defenseman Jake Gardiner scored 20 seconds later to make it 2-1.
Blake Comeau scored into an empty net with 1:11 remaining to give Colorado a 3-1 lead.