Edmonton's best opportunity in the opening frame came midway through as Eberle stole a puck deep in the Nashville zone for a breakaway but wasn't able to get Predators goalie Pekka Rinne to bite on his deke attempt, the Oilers' first shot of the game.
Nashville came close once again with a minute left in the first period as they blazed in on a three-on-one, only to see Roman Josi's shot ring off the post to keep it scoreless through the first 20 minutes.
The Oilers finally broke the deadlock eight minutes into the second period as Eberle was able to flick the rebound from a power-play point shot to Paajarvi, who hooked it into a wide-open net to make it 1-0.
The Predators tied the game on a power play of their own with just over three minutes to play in the second period as Weber blew a cannon of a point shot past Dubnyk for his fifth of the season.
The Oilers took a 2-1 lead with nine minutes remaining in the third on a short-handed goal, as Sam Gagner won a race to a lost puck to create a two-on-one, dishing the pass off to Petrell who beat Rinne with a quick one-timer.
Edmonton got an insurance goal with four minutes left as Eberle was able to beat Rinne with a high shot from the top of the circle to make it 3-1.
The Predators pulled Rinne and were able to get a goal from Josi with 8.4 seconds left, but the clock ran out on them from there.
The Predators wrap up a four-game trip on Tuesday in Columbus. The Oilers next play host to the San Jose Sharks on Wednesday.
Notes: It was the second of three meetings between the two teams this season. Nashville humbled the Oilers 6-0 in their only other game this season on March 8 ... Rinne came into the game having been yanked in both of his previous starts. Rinne was pulled in the second period of Thursday's 7-4 loss to Vancouver after allowing four goals on 12 shots faced. On Friday in Calgary he was lifted after giving up a goal on both shots he faced in the first eight minutes ... Coming into the match, the Predators had gone to overtime or a shootout in 11 of their first 28 games, tied for the highest total in the NHL ... Nashville forwards Colin Wilson, Paul Gaustad and Patric Hornqvist were all out with upper-body injuries. Defenseman Hall Gill (lower body) was also absent ... The Oilers were without forward Eric Belanger (groin), however fellow forward Ryan Jones and defenceman Mark Fistric were able to return to the lineup ... Predators defenceman Scott Hannan left the game with an upper-body injury.