Jake Evans scored shorthanded on a 2-on-1 at 9:17 to make it 4-0. Evans broke out with Paul Byron, shot and scored on his own rebound.
Tatar scored on a breakaway at 9:47 of the third period for a 5-0 lead.
"They were definitely quicker than us. Playing three games in 3 1/2 days is a lot coming out the gate," McDavid said of the Oilers, who split home games with the Vancouver Canucks on Wednesday and Thursday. "But not making any excuses for ourselves, we have to be better, we have to win more battles. They were the more rested team and they showed that, but I don't think our battle level was good enough."
Koekkoek ended the shutout at 12:59, scoring from the left circle on his blocked shot for the 5-1 final.
"It could have been a different game if we don't have him back there," Gallagher said of Price, who has 48 NHL shutouts but none against Edmonton in 14 games. "Fortunately enough, we have a great one between the pipes to make those saves. He's so calm and so cool he allows you to keep momentum on your side, and we're pretty fortunate to play in front of him every single night."
The teams play again here Monday in the third of six straight road games for Montreal to start the season.
"We park this win," Canadiens coach Claude Julien said. "We take the two points because they're important, but we have to realize we'll have a tougher opponent in two days from now."