"It was very important to finish the road trip with two points to make it somewhat acceptable," Eller said.
Mike Smith made 33 saves for the Coyotes, who have lost 10 of 11 and 12 of 14.
Arizona was shut out for the ninth time this season, one fewer than Buffalo’s NHL-worst 10.
"It was a good goalie duel," Arizona coach Dave Tippett said. "We started slow and they started fast, and Smitty held us in there. Unfortunately we couldn’t find a chance to get us back into the game."
The Canadiens broke the scoreless deadlock at 4:48 of the second period. Smith-Pelly skated down the right side to the back line and backhanded a pass to Eller in the right circle. Eller spun toward the net and beat Smith through traffic for his 10th goal.
"That’s why (Smith) didn’t get it," Eller said. "He didn’t see it."
Gallagher added an empty-net goal with 1.5 seconds left.
That was more than enough for Price, who was rarely tested over the first two periods en route to his 32nd NHL shutout and second in four starts.
Arizona managed only five shots in the first period and 13 through 40 minutes.
Price’s most difficult save came 2:35 into the third period when he dropped into the splits to make a point-blank pad stop on David Moss. The puck ended up on top of Price’s right pad, where he covered it with his glove before being pushed into the net.
"He’s obviously one of the top goalies in the league for a reason," Moss said. "He played well, especially in the third there when we made a little bit of a push."
NOTES: Price is 7-0 against the Coyotes. … Price has allowed two goals or fewer in a league-high 24 games this season. … Smith has won consecutive games only once this season. … Eller’s goal was his first since Feb. 7, a span of 15 games. … Coyotes captain Shane Doan played in his 1,379th game, moving him into 39th place on the NHL career list. … Arizona has won only seven of 28 games since Jan. 1.