The Coyotes (10-28-9) have lost five in a row.
"I thought the first five, 10 minutes we were fine," Coyotes coach Rick Tocchet said. "I thought we did all right, and then they made a push and I was a little nervous with our reaction. And then in the second and the third, I liked the way that we reacted."
Ellis tied the game 2-2 with 2:36 left in the third period. Roman Josi passed to Ellis at the point, and his one-timer beat Coyotes goaltender Antti Raanta for his second goal of the season.
"I felt me and [Josi] had some good [offensive] zone time," Ellis said. "We were moving the puck a lot. Whether we weren't getting our shots through or weren't getting a bounce here or there, I thought the whole game we created a lot of stuff up ice. Obviously, it was a big time in the game, so it was nice to get something there."
Brendan Perlini gave the Coyotes a 2-1 lead at 13:40 of the third period on a backhand from the right face-off circle. He got around Ellis and beat Pekka Rinne on the blocker side for his 11th goal.
Scott Hartnell gave the Predators a 1-0 lead at 18:17 of the second period on a breakaway. Ellis gave Hartnell a lead pass into the offensive zone, and he beat Raanta for his seventh goal.