"It's always great when you can pop two quickly in like that," Oilers forward Ryan Potulny said. "Everyone feeds off that. It gives everyone a little bit more energy, a little bit more jump in our step."
The Coyotes have dropped two of three and are yet to win consecutive games this month.
"We didn't do the little things that have been bringing us success," Phoenix defenseman Keith Yandle said. "We have to keep it simple and stay out of the box. We started off playing our simple road game and then we fell off the track a little bit and it showed."
Moreau blew past Phoenix defender Sami Lepisto and beat goalie Ilya Bryzgalov at 7:59 of the first period. Edmonton then quickly opened a three-goal cushion when Souray scored at 11:07 and Hemsky at 11:14.
Edmonton then came roaring in off the ensuing faceoff as Hemsky took a feed from Dustin Penner and deked Bryzgalov for his 100th career goal.
"That was tough, giving up a bad goal like that and then seeing them score so soon afterwards," Yandle said. "It really gave them all the momentum."
Deslauriers, getting his third straight start with regular starter Nikolai Khabibulin out with a sore back, made 15 saves in the third to preserve the shutout.
"It feels great, so exciting," he said. "It's nice to be in that club now. It's a good win for this team, too."
Deslauriers bounced back from a somewhat shaky showing against Chicago on Saturday.
"The other night he looked nervous, and stuff bounced off him, but he got by whatever jitters he must have felt before," Oilers coach Pat Quinn said. "He had really good focus tonight."
Deslauriers was able to keep it a 3-0 game after the first with a huge kick save on Scottie Upshall with about two minutes to play.
The Oilers made it 4-0 on a power play when a point shot by Souray was tipped by Horcoff with 0.1 seconds remaining in the second.
Quinn was happy with the performance as a whole, at least with the exception of the late stages when Deslauriers was frequently tested.
"We started to move the puck with authority and our guys had some jump," he said. "Our forwards were available and wanted the puck. That made for a good early performance until we got fat and happy.
"I don't know why this team would ever have that happen to them because we haven't accomplished squat yet."
Ladislav Smid kept Deslauriers' shutout bid intact four minutes into the third when he was able to sweep a puck that was sitting on the goal line out of trouble.
Phoenix outshot Edmonton 36-25.
The Oilers called up Devan Dubnyk from the AHL to serve as Deslauriers' backup. ... F Ryan Stone (knee) skated for the first time in a month. ... Phoenix defenseman Ed Jovanovski, who has missed the last eight games because of a lower-body injury, accompanied the team on its two-game trip to Alberta. The Coyotes visit Calgary on Wednesday. ... Coyotes forward Petr Prucha has gone a month without a goal.