The game was scoreless until Brashear scored 2:51 into the third period. Matt Bradley skated from behind the net and directed a pass to Brooks Laich, whose shot from the left circle was tipped past Jose Theodore by Brashear.
With 3:59 left, Steckel tapped the puck off a Colorado defender and past an out-of-position Theodore, who made 21 saves, for a 2-0 lead.
It was Steckel's fourth goal of the season.
The Avalanche, which was in danger of being shut out for the second straight game for the first time since November 2001, scored just 38 seconds after Steckel's goal on Marek Svatos' breakaway to make it 2-1.
Colorado, which had 19 shots in its 1-0 loss to the Detroit Red Wings on Tuesday night, had only nine in the first two periods and had no choice but to play aggressively.
In the final minute, the Avalanche pulled Theodore, and Tyler Arneson, who had missed the previous 12 games with a wrist injury, had a clean shot at the tie, but Kolzig kicked it away.
"I held my ground and was able to get my toe on it," Kolzig said.