Staal scored 2:55 into the third period and Ward made 34 saves to lead Carolina to a 2-1 victory over the Canadiens on Wednesday night.
Staal broke a tie with his 24th goal, beating Carey Price from the right edge of the crease. Sutter drew the Hurricanes even at 1 when he got credit for his 20th goal early in the second.
Ward stopped 11 shots in the first, 13 in the second and 10 in the third to win his second straight start after missing 17 games because of a back injury.
"We were trying a whole new thing, to see if we could win a game without actually taking a stride in the first," Hurricanes coach Paul Maurice said. "It was a short experiment -- it wasn't going particularly well. We tried first gear in the second period, but all joking aside, we were pretty good in the third."
Carolina, which has won three of four, is 11th in the Eastern Conference with 75 points, seven behind Montreal.
"You never can say never and obviously the standings don't look like they're in our favor but mathematically we're not out of it and you've got to come to the rink and compete," Ward said. "At the end of the day this is what we get paid to do and you've got to wear that jersey with some pride and compete."
Marc-Andre Bergeron scored his first goal since Dec. 31 on a power play in the first as the Canadiens outshot Carolina 12-2 in the opening period and 35-27 overall.
"We had so many scoring chances but we didn't score," center Tomas Plekanec said. "It should have been 2-0 or 3-0, and the game would have been completely different. After that we started pushing it, and it doesn't happen when you're pushing and pressing things.
Price stopped 25 shots for Montreal, which is 1-3-2 after its season-high six-game winning streak. The Canadiens are in a three-way tie for sixth in the East with the Philadelphia Flyers and Boston Bruins with 82 points, just two points ahead of the ninth-place Atlanta Thrashers.
"We were ready to play, we all knew the importance of this game going into it," Price said. "For the majority of the game we played really well. We carried the game, we just didn't get rewarded for it."
Veteran referee Kerry Fraser called his final regular-season game in Montreal, though it was his partner, Dean Morton, who was front and center on a pair of significant rulings.
Kostopoulos was sent off for goalie interference on the play and Montreal found itself with a five-on-three advantage for 40 seconds when Hurricanes defenseman Jay Harrison was called for hooking Brian Gionta on a partial breakaway that Ward thwarted with a sprawling stop at 7:18.
Carolina killed off the rest of Kostopoulos' penalty before Bergeron drove a slap shot from the point past Ward for his 11th goal at 8:04.
Sutter got credit for drawing the Hurricanes even 3:44 into the second. Morton initially waved off the goal but a video review overturned his ruling. Price stopped Sutter's initial shot -- Carolina's third of the game -- but Canadiens defenseman Roman Hamrlik slid into the rebound and pushed the puck across the goal line before the net was dislodged.
"Hammer was just doing what he needed to do, he was trying to help me out," Price said. "I make a save and it just happens."
Canadian Olympic medalists Alexandre Bilodeau and Jennifer Heil were greeted by standing ovations when they were introduced before the game. Bilodeau won Canada's first Olympic gold medal on home soil when he claimed the men's moguls title in Vancouver. Heil was second in the women's moguls after claiming gold in the event four years earlier in Torino. ... Canadiens right winger Sergei Kostitsyn left the game after his first shift of the second period because of a lower body injury. He did not return.