Nashville outshot Arizona 39-25, but Predators coach Peter Laviolette said he was "still burning" about how they finished the game.
"We put ourselves in position where we took a penalty and then played a really poor overtime," Laviolette said. "We didn't think at all through the overtime, so the ending sticks out."
Duclair gave the Coyotes a 1-0 lead at 12:41 of the second period when he charged to the front of the net out of the right corner and slid the puck inside the far post.
Duclair has two goals in 16 games since he had a hat trick in a 3-2 win at the Ottawa Senators on Nov. 18.
Ellis tied it 1-1 with nine seconds left in the second. Nick Bonino beat Stepan on a face-off to the left of Raanta, and Scott Hartnell pushed it to Ellis for a wrist shot from the top of the circle.
The Predators took a 2-1 lead eight seconds into the third period. Smith intercepted Oliver Ekman-Larsson's chip up the boards in the Coyotes zone and beat Raanta high to the glove side.
Cousins said he was surprised to be on the ice for the game-tying goal.
"I snuck out on the power play. I still don't know how," said Cousins, who is not a part of Arizona's regular power-play rotation. "Sort of right place, right time. It's nice to contribute. Obviously, it's a big goal there."
The Coyotes are 2-4-0 in their past six games, 3-10-1 in their past 14.
Nashville is 0-1-1 on a three-game road trip that ends Saturday against the Los Angeles Kings.
"We lost to the worst team in the League right now, and that's obviously not good," forward Ryan Johansen said. "There is a lot of things we've got to do better. We can definitely use this one moving forward, especially in that game Saturday. It will light a fire under our butts."
