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 2½ 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 last 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' line-up 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."
NOTES: 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.