The Canadiens, playing in their second game in as many days, however, got a good performance in the end from their backup goaltender as he filled in for an injured Olympic gold medalist.
"He battled back in the second and third and in the shootout he was outstanding," Canadiens coach Michel Therrien said of Budaj.
Desharnais, the third shooter for Montreal, slammed on the brakes at the top of the crease before sliding the puck by Marc-Andre Fleury.
"I would like to have some goals back, but we scored five goals against Pittsburgh," Budaj said. "We have to learn from the mistakes we made. We have to keep going and getting better."
Daniel Briere scored twice, once on the power play, and Max Pacioretty also scored with the man advantage, his 27th of the season. Brendan Gallagher netted his 15th for the Canadiens and Alexei Emelin also scored his second of the season.
Budaj, who stopped 27 shots, made his second successive start after turning aside 28 shots during an overtime loss one day earlier against Detroit.
"You can tell the bench guys wanted to battle back for him because he certainly worked hard," Therrien said.
He filled in for Price, who was named the top goaltender at the Olympic games after going 5-0 with a 0.59 goals-against average and two shutouts, leading Team Canada to a gold medal. Price, who aggravated a lower-body injury sustained in Sochi, will be re-evaluated Friday.
"Credit to the team because they stuck with it, they battled for me and we got the two points," Budaj said.
Crosby, who leads the NHL with 80 points, scored his 29th and added an assist. Neal scored his 21st for his 300th career point and also had an assist, while Malkin provided two assists, both extending six-game point streaks.
Olli Maatta scored his seventh of the season on the power play and had an assist, while Brandon Sutter had a short-handed goal, his 10th of the year. Deryk Engelland also scored his career-high fifth goal.
"We worked hard," Crosby said. "It was a little sloppy, and a lot of special teams. ... Probably a tough first one for everybody to get into."
The teams were tied at 3 after two periods and both scored twice in the third, sending the game to overtime.
Three of the four third-period goals were scored by special teams units, Sutter giving the Penguins a 4-3 lead with a short-handed breakaway goal and Crosby later putting Pittsburgh ahead 5-4 on the power play.
Emelin tied it 24 seconds after Sutter gave the Penguins the initial lead and Briere sent the game to overtime with his second goal of the game.
With the Canadiens trailing, the Penguins' Tanner Glass was issued a five-minute major for a hit on Emelin. The call was elbowing, but video replays showed Emelin's stick came up and hit him in the head.
"That's why I kind of followed through the way I did, to get his stick away from my face," Glass said. "It's a hockey play, it happens. I can understand why he made the call."
The Canadiens cashed in anyway, Briere tying it with his second of the game when he feathered a back-door rebound behind Fleury, who made 24 saves.
Penguins' D Kris Letang spoke with reporters Thursday morning, the first time since suffering a stroke and said he will likely play hockey again, but it's unclear if he will return this season. ... Paul Martin, out 4-to-6 weeks after breaking his hand in the Olympics, was scratched along with recently recalled D Simon Despres. ... Price, C Ryan White, and D Francis Bouillon were scratched for Montreal. ... Penguins F Jussi Jokinen recorded his 400th career point.