Captain Shane Doan scored the only goal with a backhand shot late in the first period, following an assist by Mikkel Boedker, for third-seeded Phoenix to take a commanding 3-1 lead in the best-of-seven series.
Coyotes goaltender Mike Smith delivered a flawless performance with 25 saves while Pekka Rinne made 23 stops for the fourth-seeded Predators in a losing cause.
"Coyote ugly," Smith jested to reporters after recording his second shutout of the playoffs. "We just found a way to grind out a win on the road."
Nashville played a second straight game without forwards Alexander Radulov and Andrei Kostitsyn, though this time they were left out of the lineup by Barry Trotz, and their absence may have been costly.
The home team had several scoring chances, especially during a dominant second period, but they failed to take advantage as the Coyotes improved to 4-1 on the road in the postseason.
Nashville came close to scoring late in the third period after a wild scramble in front of Smith but the official behind the net immediately indicated a no-goal, which was confirmed soon after by a video review.
"A win like this is exciting because our goaltender is so good," Doan said of Smith. "We know that we go as far as he takes us and he has been unbelievable. This solidifies how important he is and how good he is.
"I think we had the puck a little bit more tonight. We were holding on to it and we were trying to make plays. We're slow learners but hopefully we are learning our lesson. We've got to play a little bit better with the puck when we've got a lead."
Game Five is in Phoenix on Monday when a home win would earn the Coyotes a spot in the conference finals for the first time.