"I didn't really settle in until a little bit later in the game," Kinkaid said. "The shootout was a peak point and I just wanted to get the win for the guys."
Nico Hischier scored on his 20th birthday for the Devils (16-17-7), who won for the fifth time in seven games. Kyle Palmieri and Drew Stafford, in his 800th NHL game, scored in the shootout.
WATCH: [All Devils vs. Coyotes highlights]
"We had a goalie come in in relief and he was ready to play, and that's what it takes to win," Palmieri said.
Conor Garland and Christian Fischer scored, and Darcy Kuemper made 33 saves, including four difficult ones in overtime for the Coyotes (17-21-3), who lost their third in a row and fourth in five games (1-3-1).
"He gave us a chance to win, and that's all you can ask," Fischer said. "The goalies are giving us chances to win, they're making unbelievable stops. It'd be nice to reward them with a win once in a while."
Hischier gave the Devils a 1-0 lead 1:05 into the first period off a 2-on-1 rush with Palmieri. Coyotes forward Clayton Keller slid into the net and dislodged it moments before Hischier's shot went in, but the goal was awarded upon video review because it was judged the puck would have gone in had the net remained in place.