Zack Stortini and Marc Pouliot also scored for Edmonton, which ended a three-game winless streak (0-2-1) and improved to 5-5-3 at home.
Oilers coach Craig MacTavish was relieved to see his team come back after allowing Phoenix to come back and tie the score in the second period after Edmonton led 2-0.
"That game ended a whole lot better than it could have," MacTavish said. "We lost the momentum in that game in the second. We didn't have a great second but we rallied in the third and were pretty decent there."
The Oilers were just happy to head into the holidays without another bad loss.
"When you lose these kind of games, especially with the situation we are in, it can kind of fester when you have a few days off," Souray said. "Not that we are out of the woods yet by any means, but to have a couple days to get away on a win and not have to think about it. That's good timing for us."
MacTavish said he wanted the team to feel like they are coming back from the holidays with a fresh start to the season.
"I don't want them thinking about hockey," MacTavish said. "I want them to get away from the game and enjoy a couple days with their families. Maybe get on a bike too, though.
"We need a break. It has been a pretty tumultuous start."
Martin Hanzal and Enver Lisin scored for the Coyotes, who snapped a three-game winning streak. Phoenix was also unbeaten in regulation in its previous six games (4-0-2).
"We weren't good enough in the first," Coyotes captain Shane Doan said. "Obviously they jumped on us and because of that it put us in the hole and that's where we lost the game, unfortunately."
The third-period power play opportunities didn't help either.
"You give them three 5-on-3s and it's going to hurt you and obviously tonight it did," Doan said.
Stortini started the Oilers off just 5 minutes into the game. After leveling Todd Fedoruk on an open-ice hit, Stortini got his first goal of the season when he scored after a prolonged scramble in front of the Phoenix net.
Pouliot made it 2-0 less than 2 minutes later, taking a pass from Cogliano on a give-and-go and sending a shot past Tellqvist's outstretched glove.
The Coyotes tied the score in the second period when they outshot the Oilers 17-5.
Hanzal pulled Phoenix within one with 7:37 left in the period, sliding the puck past Oilers goalie Dwayne Roloson while being defended by Souray.
Phoenix tied the score on a power play with 2:23 left as Ed Jovanovski stole the puck from Steve Staios and sent it to Lisin, who blasted it from the slot past Roloson.
Phoenix was without F Peter Mueller, out because of a sore leg. ... Edmonton won all four meetings between the two teams last season, including two at home in which the Oilers gave up a combined 13 goals. .... It was just the Oilers' fifth win in 13 home games this season. Only Tampa Bay has fewer home victories.
Copyright by STATS LLC and The Associated Press
Arena: Rexall Place Location: Edmonton, Alberta
Referees: Dean Morton, Brad Meier Linesmen: Lyle Seitz, Anthony Sericolo
Attendance: 16,839 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Zack Stortini (1)
Assists: Jason Strudwick, Kyle Brodziak
Marc-Antoine Pouliot (5)
Assists: Andrew Cogliano, Ethan Moreau
Martin Hanzal (9)
Assists: Kurt Sauer, Shane Doan
Enver Lisin (6)
(Power Play) Assists: Ed Jovanovski, Olli Jokinen
Sheldon Souray (9)
(Power Play) Assists: Lubomir Visnovsky, Ales Hemsky
Andrew Cogliano (9)
(Power Play) Assists: Ales Hemsky, Lubomir Visnovsky