Pysyk, called up earlier in the day from Rochester in the AHL, scored when his deflected shot hit Martin Hanzal and became lodged in the back of Mike Smith's jersey. The goalie retreated into his crease, unwittingly carrying the puck across the line and ending the game.
Tyler Ennis also scored for Buffalo, which is 4-0-1 in its last five home games after starting the season 3-12-1 at First Niagara Center.
The Sabres started the game with just 16 skaters after a virus left Linus Omark, Cody Hodgson, Ville Leino, Marcus Foligno and Alexander Sulzer unable to play. Buffalo was only able to recall defenseman Pysyk and left wing Johan Larsson from Rochester because the team en route to the Spengler Cup tournament in Davos, Switzerland.
Hanzal scored for the Coyotes, who concluded a four-game road trip with a 1-1-2 record. Smith made 30 saves.
Buffalo was outshot 15-3 in the first period, though the two best chances belonged to the Sabres.
Phoenix 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 but it was determined Hanzal's stick struck the puck below the height of the cross bar.
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.
Zemgus 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.
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.
Jamie McBain had a partial breakaway in overtime but was stopped by Smith, who made a second stop on McBain moments later.
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.