Bjugstad, 26, and McCann, 22, made it to PPG Paints Arena minutes before the puck drop in Pittsburgh's 5-3 win over the Ottawa Senators.
They both took the morning skate with the Panthers and boarded a plane to Pittsburgh when they learned of the trade. They received a police escort and arrived at the arena before faceoff, still with their Panthers equipment bags as they hurried to join their teammates on the ice.
Bjugstad and McCann first appeared on the bench before the start of the national anthem.
"Showing up at game time was interesting and might have been better for us because there wasn't much thinking," said Bjugstad, who contributed an assist and won the offensive zone faceoff in his first game with the Penguins. "It was pretty surreal. We showed up to the rink and just played. I have dreams every once in awhile where I'm late to a game, and that's kind of what it felt like, but luckily it ended up well, so it was good.
"Usually I have my coffee and do some stretches, but I didn't have any of that. We got here, threw our gear on and ran out for the anthem."
Bjugstad, who is signed through the 2020-21 season at an average of $4.1 million per year, has five goals and seven assists in 32 games for the Panthers this season. McCann, signed through the 2019-20 season at an average of $1.25 million, has eight goals and 10 assists in 46 contests with Florida.
Panthers general manager Dale Tallon said it was difficult to part with Bjugstad and McCann.
"You've got to put the personal feelings aside and do what's right for the franchise," Tallon said. "It's hard to do sometimes, but necessary."
The deal ends Pittsburgh's experiment in attempting to make Brassard, 31, the team's third-line center. Acquired in a deal with Ottawa last season, Brassard struggled to generate much offense during his brief time with the Penguins, collecting just 12 goals in 64 games with Pittsburgh.
Sheahan, 27, brought over in a trade with Detroit early in the 2017-18 season, had 18 goals and 23 assists in 122 games with the Penguins. Brassard and Sheahan both have expiring contracts, and the Panthers -- who are trying to get back into the Eastern Conference playoff hunt -- might move them again before the Feb. 25 trading deadline.
Brassard, who has also played for the Senators, Rangers and Blue Jackets since being drafted in 2006, has 443 career points.
"We'll go game-by-game and week-by-week and then we'll decide before the deadline what we're going to do," Tallon said.
Pittsburgh also sent a second-round pick and two fourth-round picks in the 2019 draft to the Panthers to complete the deal.
The Associated Press contributed to this report.