Biron, giving Dominik Hasek a day of rest before Sunday's game at Dallas, finished with 33 saves as the Sabres beat Colorado for the third straight time in Denver.
Colorado erased a two-goal deficit in the second period but failed to break through against Biron in the third.
Dumont set up the breakaway when he took the puck away from Sakic deep in the Buffalo zone.
Roy finished with 13 saves.
Colorado, playing its first game in six days, trailed 2-0 before Forsberg scored midway through the second period. It was the Avalanche's first goal against Buffalo at home since Oct. 28, 1997 - a span covering 161 minutes, 22 seconds.
The Avalanche tied the game when Sakic scored on power play with four seconds left in the period. At that point, Colorado had a 25-6 shot advantage.
Buffalo, in the midst of a five-game road trip, took only three shots in the first period but scored on two of them.
Brown converted a centering pass from Vladimir Tsyplakov after a Colorado turnover at 3:28 and Gilmour scored a power-play goal at 18:27.
Jason Woolley set up Gilmour by drawing Roy out of position near the left post. Gilmour took the pass and put a shot between the legs of Colorado defenseman Ray Bourque, who moved in front of the net while Roy scrambled outside the crease.
©2001 The Associated Press. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed