Penguins goaltender Marc-Andre Fleury made 17 saves for the win, while Calgary counterpart Miikka Kiprusoff, who needs one win to become the franchise leader at 263, stopped 32 shots.
Crosby is skating with the team on the road trip, but he has not been cleared for contact. He played 41 games last season before suffering concussions in January that ended his season.
Crosby scored a hat trick in Pittsburgh's 4-1 win over Calgary in their previous meeting Nov. 27, 2010.
The Penguins pushed the puck quickly through the neutral zone. They were good on Saturday at getting behind Calgary's defenders and scoring from close range in their second-period burst.
"We have a pretty dangerous club when we do those things," Staal said.
Calgary led 1-0 after the first period, but it was all Pittsburgh in the second period as the visitors outshot Calgary 16-5.
After Malkin made it 4-1, tempers flared.
The Penguins took exception to a check on Matt Cooke by Flames defenseman Cory Sarich and that prompted a line rumble near Calgary's net. Malkin later threw a couple punches at Glencross and both were sent to the penalty box for roughing and slashing respectively.
Calgary scored twice in the third period to draw within a goal. With Kiprusoff pulled for an extra attacker, Staal put the puck in the empty net.
"A lot of lines played well at certain times in the game," Staal said. "I still think we could have obviously played a better full game than we did. The third period wasn't our best, but we held them off for the two points."
Calgary had nine days between its final preseason game and first of the regular season. The Flames didn't have their opening night lineup for any preseason games because of injuries.
Captain Jarome Iginla didn't play a game until Saturday because of back spasms he experienced the first day of training camp.
"The second period is definitely what cost us tonight," Iginla said. "We got sloppy, we had way too many turnovers, we didn't play good hockey.
"They're a quick team and we can't turn the puck over like that. We played the whole period in our zone because of it. In the first and third, we were solid, but we can't give any team a period like we gave them."
Jokinen scored Calgary's third goal at 16:23 of the third period. Fleury bobbled the puck behind his own net and Niklas Hagman backhanded the puck out front to the Finn, who beat Fleury stick side at 16:23.
Calgary's Bourque made it 4-2 in the third, deflecting an Alex Tanguay shot by the Pittsburgh goalie at 9:08.
Malkin tucked a rebound by Kiprusoff at 13:01 of the second period for a power-play goal. It was the Russian's first of the season although he scored during the shootout in Vancouver. Adams made it 3-1 at 10:11, tapping the puck past Kiprusoff from the slot.
Niskanen gave the Penguins a 2-1 lead at 6:40 when Calgary was serving a penalty for too many men. Niskanen batted in a rebound off Kiprusoff's outstretched pad for the defenseman's second goal as a Penguin since he was acquired from Dallas last season.
Pittsburgh caught Calgary's power play off guard when Pascal Dupuis head-manned the puck to Kennedy stepping out of the penalty box. With the Penguins rushing Calgary's zone, Kennedy opted to keep the puck and put a wrist shot between Kiprusoff's pads at 1:04.
Calgary moved the puck quickly in Pittsburgh's zone in the opening minutes to create scoring chances, but didn't get its first goal of the season until 15:32 to take a 1-0 lead into the second period.
Flames defenseman and new assistant captain Mark Giordano drove the net and passed cross-ice to Glencross, whose shot from the faceoff circle deflected off Fleury and into the net.
Pittsburgh plays its first three games away from home for just the second time in its history. The only other instance was the lockout-shortened season of 1994-95, when it defeated Tampa Bay, Florida and New York in January.
Center Roman Horak, 20, had an assist for Calgary in his NHL debut . . . Jordan Staal collected his 200th career point assisting on Niskanen's goal. ... Calgary forward Niklas Hagman played his 700th career game Saturday. ... Pittsburgh defenseman Brooks Orpik was scratched with a lower-body injury.
Copyright by STATS LLC and The Associated Press
Arena: Scotiabank Saddledome Location: Calgary, Alberta
Referees: Tim Peel, Brian Pochmara Linesmen: Lonnie Cameron, Don Henderson
Attendance: 19,289 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Curtis Glencross (1)
Assists: Mark Giordano, Scott Hannan