Jesper Bratt had two goals and an assist, Jack Hughes had three assists and Akira Schmid made 30 saves for the Devils (1-0-1). Dougie Hamilton had a goal and an assist, and Tyler Toffoli had two assists.
"It's an important point that we lost, and I thought we had some good looks in overtime that we definitely could have ended the game," Bratt said. "In the shootout, we didn't have great details and their goalie played great when we were trying to find areas to score. It's just how it is in shootouts sometimes."
Schmaltz scored a power-play goal on a one-timer from the left face-off circle to tie it 3-3 at 13:55 of the third. Cooley had the primary assist on the play when picked up a loose puck, skated to the slot and fired a backhand pass to setup Schmaltz.
"We believe in him, we know what he can do," Coyotes coach Andre Tourigny said of Cooley. "You will see him all the time as a rookie, but we see him as a player for us. He's a member of our team, an important guy on our team and he will grow every game."
Dumba made it 1-0 in his Arizona debut at 6:22 of the first period. The 29-year-old defenseman, who signed a one-year, $3.9 million contract Aug. 7, scored on a wrist shot from the right face-off circle.
Durzi, a defenseman acquired in a trade with the Los Angeles Kings on June 24, pushed the lead to 2-0 with a power-play goal at 4:23 of the second period.
"It was a disappointing start to the game," New Jersey coach Lindy Ruff said. "I thought they looked fresher, ready to play in their first game. I thought their energy was really good, and they got a step on us in most places."