Eichel scored into an empty net with 32.9 seconds to play for his eighth goal.
"It would be easy to get down and go in a different direction, but the guys fought the whole night," Sabres coach Phil Housley said. "It was a total team effort up and down the lineup. Robin had a really solid game for 60 minutes and I thought our leaders really led tonight."
MacKinnon tied the game 2-2 at 9:58 of the third period when he beat Lehner to the short side for his second of the game and 10th of the season.
"When they tied it up there we came up with a big goal, got us ahead again," Housley said.
The Sabres took a 2-1 lead in the second period on goals by Pouliot and McCabe.
Pouliot scored shorthanded at 5:32 off a pass from Ryan O'Reilly, who skated into the Avalanche end as defenseman Erik Johnson slid toward the net. O'Reilly passed across to Pouliot on the right side for a shot past Varlamov for a 1-0 lead.
The Avalanche tied the game 1-1 at 12:22 when MacKinnon beat Lehner to the short side from the left circle.