Darcy Tucker scored the tying goal with 10.7 seconds in regulation and added the winner in the fifth round of the shootout in the Avs' 3-2 victory Saturday night.
Former Oilers star Ryan Smyth also scored in regulation for Colorado. The Avalanche have won three straight following a five-game losing streak.
"We didn't play our best game tonight," Tucker said. "They played a pretty good hockey game for the most part and we just found a way to win it and sometimes you just need to find ways to win games when you haven't played your best.
"We got some positives out of a game like this. The fact that we did come back and found a way to win when we didn't play our best is obviously great for us."
"We hung around -- which was good -- and most of the reason we hung around was because our goalie made the saves," Granato said.
With the Oilers nursing a 2-1 lead late in the third period, the Avs were given a great opportunity following penalties to Fernando Pisani and Ethan Moreau. With Budaj on the bench, the Avs had a 6-on-3 advantage and made it count when Tucker redirected a point shot past Dwayne Roloson to send the game to extra time.
Colorado's Wojtek Wolski and Edmonton's Sam Gagner traded goals in the first round of the shootout. Budaj and Roloson both stopped the next three shooters they faced before Tucker gave Colorado the lead with a shot that eluded Roloson between the legs.
Shawn Horcoff, Edmonton's fifth shooter, had a chance to extend the shootout but fired a harmless shot that Budaj redirected with his right pad.
Roloson expressed frustration at his team having wasted a solid effort.
"It was unfortunate that they were able to get one late," Roloson said. "Especially because we played well in that game. That was probably one of the best games we have played defensively all year."
Edmonton struck first just shy of 7 minutes into the game. Tom Gilbert's power-play point shot struck Nilsson in front, but the forward grabbed the loose puck and hooked it past Budaj.
Colorado tied it 4 minutes later when Smyth crashed the net and slid his own rebound under Roloson while being knocked down. It was Smyth's first career goal against his former team.
The Oilers regained the lead 7 minutes into the second period when Reddox, playing in his second NHL game, scored his first NHL goal. He grabbed the puck from behind the Avalanche net and chipped it over Budaj's glove.
"It is unbelievable," Reddox said. "I was pretty zoned in. I saw it coming off the boards and just tried to lift it up. I can't even describe the feeling."
Colorado captain Joe Sakic (back) missed his second game in a row. ... Dustin Penner and Kyle Brodziak were healthy scratches for the Oilers. ... The game was the 1,000th in the regular season for Colorado since the team moved to Denver from Quebec City in 1995.