With 3:47 gone in overtime, Sabres defenseman Mark Pysyk took a shot that hit Martin Hanzal and became lodged in the back of Smith's jersey. The Coyotes goalie retreated into his crease, unknowingly carrying the puck across the line and ending the game.
"I didn't feel it at all," Smith said of the bizarre goal. "I didn't know what happened. I knew that puck went up in the air but had no idea where it went after that."
Ryan Miller made 36 saves for the Sabres and said he has never seen anything quite like Monday's ending.
"A little unlucky for Smith there," Miller said. "I think he'll be shaking his head for a while. It's a little bit of a Christmas present for us, and we'll take it."
Tyler Ennis also scored for Buffalo, which is 4-0-1 in its past five home games after starting the season 3-12-1 at First Niagara Center.
The fans gave the Sabres several standing ovations throughout the contest, and the team went to center ice to salute them after the game.
"I play on emotion, and when fans are just supportive like that, it's a huge boost," Sabres rookie center Zemgus Girgensons said. "First 30 games fans were there for us but we didn't play well."
Hanzal scored for the Coyotes, who concluded a four-game trip with a 1-1-2 record. Smith made 30 saves.
Coyotes coach Dave Tippett said the odd ending fit his team's performance.
"The game is an honest game," he said. "If you're supposed to win, usually you can win. And we didn't deserve to win tonight, so those things happen."
Phoenix outshot Buffalo 15-3 in the first period, although the two best chances belonged to the Sabres.
The Coyotes opened the scoring 19 seconds into the second period when Hanzal raised his stick to deflect Connor Murphy's slap shot over Miller. The play was reviewed, and it was determined Hanzal's stick struck the puck below the height of the crossbar.
Miller didn't like the call and directed blame to the NHL situation room in Toronto, which reviewed the goal.
"It hit the top of his blade, and his blade was up high," the goalie said. "The puck wasn't rising, it was already over the crossbar in my mind, so I think Toronto got it absolutely wrong."
Buffalo native and former Sabres left wing Tim Kennedy recorded an assist on the goal.
Miller kept the deficit at 1 by coming out to the top of his crease to stop a high shot from Michael Stone.
Girgensons was thwarted on a breakaway just two minutes later as Murphy grabbed the rookie's stick before he could get a shot on Smith. There was no penalty on the play.
Drew Stafford left the Sabres with 15 available skaters after receiving a five-minute elbowing penalty and game misconduct. Facing the boards to the right of Smith, Stafford swung his elbow backward at oncoming Oliver Ekman-Larsson and caught the defenseman in the face.
The Sabres killed off 2 1/2 minutes of the major penalty before Hanzal took a roughing penalty. In the remaining 30 seconds, Smith stopped Ott's slap shot during a 2-on-1 break.
Buffalo's penalty kill has been successful on 21 of its past 22 chances.
"Short bench, all the penalties we had, it didn't seem like anything in the game was really going our way," Girgensons said. "We just worked probably the hardest we've worked throughout the whole season and stuck to the system. That's the way we won."
Ennis tied the score 14:01 into the third. Girgensons forced a turnover at the Coyotes' blue line and found Ennis, who beat Smith for his eighth goal of the season.
"The guy just fell, and I lifted his stick and slid the puck through the two guys I was hoping it would go through, and Ennis finished it really good," Girgensons said.
Jamie McBain had a partial breakaway in overtime but was stopped by Smith, who quickly had to make a second stop on McBain.
The game ended moments later with Pysyk's goal.
"I just put it on net, figuring I'd maybe get a rebound," Pysyk said. "Just popped up right in the air and was lucky enough that he spun and it went right into the net."
Smith was embarrassed by his team's effort considering the Sabres' lineup woes and an impending three-day break for Christmas.
"That was embarrassing as far as I'm concerned," he said. "You go on a Christmas break, you have three days off and you come out like that? I think that's a bunch of garbage as far as far as I'm concerned."
The Sabres placed C Kevin Porter on injured reserve before the game and assigned D Chad Ruhwedel to AHL Rochester. ... Phoenix scratched Gilbert Brule for the first time in three games. ... John Scott played forward and defense for the Sabres.
Copyright by STATS LLC and The Associated Press
BUFFALO, NY - DECEMBER 23: Fans arrive for the game between the Buffalo Sabres and the Phoenix Coyotes on December...