Late in the third period, Ennis stuck his stick into the net and accidentally prevented teammate Luke Adam's shot from crossing the goal line.
Colorado eventually capitalized on the break as David Jones scored a power-play goal 4:17 into overtime, helping the Avalanche snap a two-game skid with a 4-3 win over the Sabres on Tuesday night.
"It might be one of the most bizarre breaks you can get -- hit your own stick to stop it from going in the net," Sabres coach Lindy Ruff said.
The Avalanche certainly weren't arguing about their good fortune.
On a night when Budaj stopped 37 shots, the save that will be remembered most won't be one of his.
"You catch some breaks, which was good," Avalanche coach Joe Sacco said.
With around 6 minutes left in regulation, Adam launched a shot from in close that appeared to miss the net altogether. Only after numerous overhead views of the play could it be seen that Ennis' stick was near the goal line and kept Adam's shot from going in.
The play was reviewed, and it was determined that the puck didn't completely cross the line.
"It was a matter of centimeters," Adam said. "It was pretty close."
Ennis said he was just trying to jam the net, and his stick got caught. He wasn't attempting to tip in the puck.
"I had bodies on me and somehow it hits my stick," Ennis said. "It's disappointing. A tough loss."
To add insult, Ennis was the player in the box when the Avalanche scored the winner in overtime after drawing a high-sticking penalty.
It was Jones' first career overtime goal.
"It's unbelievable," Jones said. "I didn't know what to do I was so excited."
The Sabres had a tough night on the power play as they went 1-for-8. Hecht had an open goal to shoot at on a 4-on-3 advantage in overtime, but the puck hit the side of the net.
"Sometimes it comes back to bite you," Ruff said.
The Avalanche exorcised some power-play demons with Jones' goal. Before that, Colorado was in an 0-for-14 drought spanning four games.
Colorado also halted a four-game winless streak at home.
"Nice to get the monkey off our back," Jones said. "It's been a few games in a row that we've kind of squandered the lead, so it's great to get the two points here."
The Sabres took a 3-2 lead early in the third period when Hecht and Montador scored 1:03 apart.
The Avalanche took a quick timeout to regroup and answered when Duchene knocked in a rebound that was loose in front.
Winnik staked the Avalanche to a 1-0 lead late in the first period by beating Miller with a wrist shot on a breakaway. T.J. Galiardi set up the goal after flipping the puck up ice to an awaiting Winnik.
The lead lasted only 33 seconds as the Sabres took advantage of a double-minor penalty on Ryan O'Byrne for high-sticking. Vanek, standing on the doorstep of the net, tapped in a pass from Drew Stafford for his team-leading 15th goal.
Buffalo then allowed a short-handed goal when Winnik sent a pass across the crease to O'Reilly, who slapped in a shot to give the Avalanche a 2-1 lead after 20 minutes.
Still, it was a game that got away from the Sabres.
"The chances are there, we had that open net, we had a chance in overtime," Adam said. "I mean, we just have to capitalize on our chances."
The Sabres played in Denver for the first time since Oct. 25, 2008, a game they lost in a shootout. ... Avs prospect D Tyson Barrie will play for Canada in the gold-medal game against Russia on Wednesday in the World Junior Championship. ... The Avs are 3-0-0 against Northeast Division teams this season. ... Fleischmann and Stafford both had two assists.