"It was a Jekyll-and-Hyde night," Coyotes coach Dave Tippett said. "A perfect ending."
Morris scored and Jordan Szwarz added his first NHL goal for Phoenix, which has points in 10 of its past 11 games and is 9-3-2 on the season.
Ten Phoenix players recorded at least a point.
"I think we all believe we all can contribute in some way or form," Morris said. "We believe in ourselves. I don't know if there's ever a good way to fall behind, but we were fortunate to make it 4-4."
Thomas Greiss made 36 saves and stopped all five shootout attempts for the Coyotes, who rallied from a 3-0 deficit.
"I think it depends on how you fall behind," Tippett said. "After the first period, it's not like we thought we were dead and out of it. The feeling was let's see if we can get a break and we'll get back into it."
Nashville was playing the first game of a franchise-record 17-day, seven-game road swing to accommodate the Country Music Awards on Nov. 6 at Bridgestone Arena.
"We shot ourselves in the foot," Nystrom said. "That can't happen. It's tough. We should have had two points in that game, 100 percent."
Gaustad gave the Predators a 4-2 lead on a rebound at 4:15 of the third before the Coyotes rallied.
Doan cut the lead to 4-3 on a power-play goal from the bottom of the left circle at 6:16.
A little more than two minutes later, Doan found a streaking Antoine Vermette skating down the slot behind the Predators' defense.
"We made too many mistakes," Nashville coach Barry Trotz said. "That's not smart hockey."
Hendricks gave the Predators a 1-0 lead at 9:00 of the first period, taking a long outlet pass from Legwand at the Phoenix blue line and scoring from the top of the right circle.
Nashville had a two-man advantage for 62 seconds near the end of the period, and Legwand took advantage, bouncing a shot from just off the goal line right of the net off the right skate of Coyotes defenseman Zbynek Michalek standing at the top of the crease.
Michalek set up Szwarz early in the second with a cross-ice pass, and the rookie, playing in his second career game, lofted a soft wrist shot toward the net that slid under Hutton's glove.
Morris cut the lead to 3-2 at 9:51 of the period on a rising slap shot from the top of the right circle.
Doan nearly tied the game with six minutes left in the period, slipping a shot between Hutton's pads, but Viktor Stalberg reached behind the goalie and cleared the puck inches off the goal line.
Hutton finished with 21 saves.
Tippett tied Bobby Francis for first place on the franchise wins list with 156. ... Phoenix is one of four teams in the league that have not lost at home in regulation (6-0-1). ... Doan has three goals and an assist in five career games on Halloween. ... Nashville is unbeaten in regulation when scoring first (5-0-2). ... Hendricks left the game 33 seconds into the second period with an upper-body injury after a collision with Morris. ... Coyotes C Jeff Halpern missed his second straight game with an upper-body injury. ... Legwand has seven points -- a goal and six assists -- in his past six games. ... Phoenix has been outscored 17-5 in the first period. ... Szwarz was recalled from Portland of the American Hockey League on Tuesday. ... Phoenix leads the league with 40 points scored by defensemen.