Evander Kane scored a tiebreaking goal in the third period and Robin Lehner had 35 saves and for the Sabres (7-17-4).
Nathan MacKinnon scored two power-play goals, Alexander Kerfoot had two assists and Semyon Varlamov stopped 31 shots for Colorado. The Avalanche (12-12-2) have lost three straight and dropped four of five on their homestand.
Buffalo has scored 60 goals this season, the fewest in the NHL and an average of 2.14 per game, and got a break on their second goal of the game.
McCabe beat Varlamov with a screened shot at 18:59 of the second, but the Avalanche challenged the play was offsides. Replays showed the puck clearing the zone with two Buffalo players still below the blue line as Johan Larsson tipped it into the offensive zone.
After a review the goal was upheld, Colorado was assessed a delay-of-game minor, and the Sabres took a 2-1 lead into the second intermission.
The NHL Situation Room sent out a tweet explaining the ruling:
"...All Sabres players tagged up at the blue line prior to Johan Larsson touching the puck in the offensive zone, rendering the play onside."
MacKinnon tied it with this 10th goal midway through the third period but Kane got the go-ahead goal, his 13th, less than two minutes later to give the Sabres the lead.
Eichel scored into an empty net with 33 seconds left to ice it.
The Sabres scored first when Ryan O'Reilly, who had two assists, stole the puck on a Colorado power play and fed Pouliot for his seventh goal of the season at 5:32 of the second period.
MacKinnon tied it at 1-1 when he scored on Colorado' fourth power-play chance on a shot to the short side at 12:22 of the second.
NOTES: Buffalo C Scott Wilson had 14:19 of ice time a day after coming to the Sabres in a trade with the Detroit Red Wings. ... Colorado LW Gabriel Landeskog served the last contest of his four-game suspension for his crosscheck of Calgary's Matthew Tkachuk on Nov. 25. ... The Sabres waived F Matt Moulson on Monday, and the veteran cleared waivers on Tuesday. He is on loan to the Los Angeles Kings' AHL affiliate. ... Colorado recalled F Rocco Grimaldi from San Antonio of the AHL on Monday. It is his second call-up of the season.