"I was worried after two periods that they didn't have a lot of shots," Trotz said. "I knew they'd have their best effort in the third, come with everything and that we'd have to weather some storms. Pekka was up to the challenge."
Rinne stayed focused by playing the puck whenever it came near him, infrequent as the chances may have been.
"Obviously it's hard when you only face eight shots in the first two periods," Rinne said. "But it's all mental. You just try to be sharp. They didn't have anything the first two periods and the guys showed how important a game it is. It's a life or die game for us."
The Predators snapped a three-game losing streak, and moved into a tie for eighth place in the Western Conference with Edmonton -- a point behind Dallas.
Ilya Bryzgalov made 31 saves for the Coyotes, who have lost five straight and eight of their last nine. Phoenix is 4-16-1 since the All-Star break.
"I thought it might have been his best game of the year tonight and he's had a few good ones," Coyotes coach Wayne Gretzky said.
Dumont opened the scoring with 5:59 left in the second period, one-timing Steve Sullivan's pass over Bryzgalov's right shoulder from the high slot.
Legwand added an empty-netter with 30.3 seconds left, skating in from center ice and flipping the puck into the net from the left faceoff circle.
Rinne rarely was tested, facing only four shots each in the first and second periods. The Coyotes managed 10 shots in the final period, but had almost no real scoring chances.
"The problem is it was all in the third period," Coyotes captain Shane Doan said. "Eight shots after two periods in your home building, it's embarrassing. It's going to be better moving forward. It will be better."
Rinne is 12-3-1 since Feb. 1 and recorded his third shutout in six weeks. Three of those wins have come against the Coyotes, with Rinne stopping 72 of 74 shots.
"We all knew Pekka was a solid goaltender but what he's done this year has been pretty amazing," Dumont said. "We have two good goaltenders and we know every time they make a big save we have to score big goals for them."
The Predators were without center Jason Arnott and Radek Bonk. Arnott has missed consecutive games with an upper-body injury, while Bonk has been out of the lineup since Feb. 22, also because of an upper-body injury.
The Predators won the final three games of the four-game series against the Coyotes. ... Phoenix has scored two goals or fewer in nine straight games. ... Dumont, Nashville's leading scorer, has goals in consecutive games after being held without a point in the Predators' previous three contests. ... Six Coyotes players acquired during the team's five-game trip that started just before the trade deadline made their home debuts.