WATCH: All Ducks vs. Coyotes highlights
What happened was, the knob on Smith's stick broke and he failed to get the puck out of the crease.
"The way things are going right now, my stick breaks, I didn't get anything on it, and it ends up in the back of the net," Smith said. "I tried to do too much."
Anaheim (20-12-8) won its third straight and got its first win in eight overtime games this season.
Joseph Cramarossa and Chris Wagner scored for the Ducks. John Gibson made 21 saves.
Kase had a point in consecutive games for the first time in his 25-game NHL career and has three goals in his past six games.
"If you look at the game as a whole for him, he had four quality scoring chances and he was around the puck all night," Ducks coach Randy Carlyle said.
Martin Hanzal and Oliver Ekman-Larsson scored for the Coyotes (11-22-6), who lost their ninth straight game. Smith made 32 saves.
"We battled hard," Coyotes coach Dave Tippett said. "We played with an unbelievable amount of desperation and hung in there. It's too bad we couldn't get the win."
Cramarossa deflected Corey Perry's wrist shot past Smith at 6:35 of the first period to make it 1-0. Hanzal tied it 1-1 1:19 into the second.
Wagner, who was in the American Hockey League for 13 of the Ducks' past 14 games, scored his first NHL goal since Oct. 25 at 16:05 of the second right off the faceoff to put Anaheim ahead 2-1.
Ekman-Larsson tied it 2-2 with a power-play goal 3:12 into the third.
The Coyotes lost three players in the third period. Forward Jordan Martinook (upper body) did not play, and Hanzal (lower body) and defenseman Jacob Chychrun (upper body) left the game.
The Ducks played a second straight game without captain Ryan Getzlaf (lower body).