Sam Gagner also scored in the shootout and Teemu Hartikainen had the other goal in regulation for the Oilers, who rallied from a two-goal deficit to close out their five-game homestand 2-3.
Kyle Chipchura and Mikkel Boedker scored for the Coyotes, who had a two-game winning streak snapped. Phoenix goaltender Mike Smith was excellent with 41 saves, but couldn't stop Eberle and Gagner in the shootout.
The Coyotes started the scoring after some prolonged pressure when Chipchura shoveled a shot toward the net that just trickled across the goal line before Oilers starter Nikolai Khabibulin could reach back to glove it.
Edmonton had a couple of opportunities to tie the game up late in the first, both short-handed. Eric Belanger and then Gagner were able to break loose on breakaways, but both were stoned by Smith.
The Oilers came close again five minutes into the second frame as Ryan Jones lifted a rebound over Smith, only to see defenseman Keith Yandle swoop in and glove it out of harm's way.
Phoenix made it 2-0 midway through the second with Nick Schultz serving a double minor for high-sticking. Edmonton had killed off the bulk of the penalty before a Mikkel Boedker shot got past Khabibulin and hit the post, allowing Lombardi to sweep in the rebound.
Edmonton finally got on the board with 3:38 remaining in the second as Nail Yakupov showed some speed hustling for a dump in and send a backhand pass to Eberle who chipped it past Smith.
The Oilers tied the game eight minutes into the third on the power play as Ryan Smyth tipped Justin Schultz's shot and it went straight to Hartikainen at the side of the net for his first of the season.
Edmonton had the best chance in overtime with a power play, but Smith was able to make a dramatic glove save on a hard shot by Gagner.
Khabibulin stopped Boedker and David Rundblad in the shootout and finished with 34 saves.
The Oilers now embark on a nine-game road trip, starting Monday in Chicago, with the Canadian men's curling championships occupying Rexall Place for the next two weeks.
Notes: Edmonton forward Taylor Hall served the first of a two-game suspension for his hit on Minnesota's Cal Clutterbuck on Thursday. ... Jones played his first game of the season since suffering an eye injury during a practice session during the lockout. ... Edmonton D Ryan Whitney was a healthy scratch for the fifth time this season. ... Phoenix was without F Radim Vrbata (lower body) and D Derek Morris (upper body).