Kostitsyn had his first career hat trick and Price stopped 32 shots, leading the Montreal Canadiens to a 3-2 victory.
"I liked our effort tonight and thought we deserved better," Penguins coach Michel Therrien said. "There are going to be nights when the result is not on your side. Tonight was one of those nights. Carey Price was outstanding and stole the game."
Tomas Plekanec added two assists for Montreal, 3-0-1 in its past four games.
Pittsburgh held a 34-19 advantage in shots on goal and had five power plays to the Canadiens' one. Kostitsyn scored the winner 2:21 into the third, a period in which the Penguins had three power plays and 17 shots. But they could not beat Price for the equalizer.
"When you have an offense like the Penguins have, and especially in the third period when we gave them three power plays, it's never easy," Montreal coach Guy Carbonneau said. "I think Carey's always been known as a guy who can finish. I think tonight he was ready."
Earlier in the day, Kostitsyn was questionable. He missed the previous two games with a minor leg injury and said during the morning skate that his availability would be a game-time decision.
But Kostitsyn came out strong, scoring twice in the first period.
"I just went to the net and (Plekanec) made me two good passes and I scored," he said.
Kostitsyn, who entered the game with six goals this season and didn't have one since Dec. 4, scored 1:15 in on a pass from Plekanec after Josh Gorges' shot went wide but bounced hard off the boards.
Kostitsyn's second goal came with only 1.6 seconds to play in the first, giving the Canadiens a 2-1 lead after Dupuis had tied it four minutes earlier. Kostitsyn scored on a one-timer from the slot following Plekanec's pass from behind the net.
"We want (Kostitsyn) to shoot," Carbonneau said. "I think he has the best shot on our club. He should be ready to shoot from everywhere, and tonight that's all he did. He got good shots right on the net and turned them into goals."
"I know we lost, but we had a pretty good effort," Crosby said. "We didn't give them much. We had a couple mistakes on the first two goals and the third goal was a pretty good shot. ... They really didn't get anything after that. We had a lot of chances to tie it up and even take the lead. Our effort was there."
Crosby tied it at 2 early in the second period by banking the puck off Price while he was attempting to tuck it in on a wraparound attempt.
"I think we deserved to win tonight," Dupuis said. "The big thing is the whole team battled hard tonight. Unfortunately we didn't get the win, but if we work like that every night I think we'll give ourselves a chance to win."
Pittsburgh controlled play during the third period but could not tie it, partially due to some quality saves by Price, who has helped Montreal earn at least a point in each of his past 10 starts.
"The guys did a pretty good job of keeping all of the rebounds out of their reach," Price said. "That makes a heck of a difference for me."
Kostitsyn had two previous two-goal games. ... Penguins D Ryan Whitney assisted on Dupuis' goal for his first point of the season. Whitney was playing in his third game since offseason foot surgery. ... For the past nine games, Pittsburgh has alternated wins and losses.