Nashville played without leading scorer Filip Forsberg, who was placed in COVID-19 protocols just before the game.
The Predators were just fine without him against the last-place Coyotes, scoring three first-period goals in a chippy game that featured 25 penalties for 38 minutes.
Forsberg had scored four goals in three games and had multiple points in each, but was a last-minute scratch.
Nashville rolled right on without him, scoring 56 seconds in when Ekholm punched in a rebound and fell over Vejmelka after an odd-man rush.
Then things got testy.
Coyotes center Liam O'Brien dropped gloves with Mark Borowiecki two seconds after the opening goal, then Arizona's Phil Kessel chopped at Ekholm for slashing him. Arizona's Dysin Mayo also got into a wrestling match against the boards with Matt Luff.
And that was in the first 6 1/2 minutes.
The penalties kept coming — 12 in the first period — and Trenin scored on a 4-on-4 with a shot from the right circle Vejmelka couldn't see through traffic. Jeannot made it 3-0 by charging to the net and one-timing a pass from Matt Duchene.
The pace of goals (none) and penalties (two) slowed considerably in the second period. Saros continued to be sharp, making a glove save on Travis Boyd's short breakaway and a couple others on odd-man rushes.
The testy tempo picked back up to start the third period.
Keller scored early in the period off a faceoff, lifting a sharp-angle shot that hit the crossbar and the back of Saros' helmet.
Andrew Ladd and Trenin fought 25 seconds later after Trenin went after Keller. O'Brien went back to the penalty box for roughing and Josi scored on the ensuing power play, beating Vejmelka to the glove side through a screen.
Nashville: Hosts Colorado on Tuesday.
Arizona: Hosts Toronto on Wednesday.
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John Marshall, The Associated Press