The linemates each had two points, with the 20-year-old Gallagher scoring his second goal of the season, as the Canadiens stretched their winning streak to four games with a 4-3 victory over the Winnipeg Jets on Tuesday.
"There's always going to be pressure on me, but you don't want to take your foot off the pedal," said the stocky, 5-foot-8 Gallagher, who has four points in as many NHL games.
"You want to play the same way and show the coaches that you're not going to relax just because they gave you the good news."
The 18-year-old Galchenyuk, the third overall pick in the June draft, had a pair of assists and has five points in five games to start his NHL career.
Tomas Plekanec's tiebreaking power-play goal at 5:31 of the third period was the game-winner for Montreal.
Nik Antropov and Blake Wheeler also scored for Winnipeg, which ended a three-game winning streak.
First-year general manager Marc Bergevin ended debate over whether Galchenyuk should be sent back to his junior team and Gallagher to the AHL when it was announced Monday the two had made the team.
Gallagher, a fifth-round pick in 2010, responded with a performance that earned him first star honors and prolonged applause from the 21,273 Bell Centre fans.
Galchenyuk was selected as the second star.
"As a line, we just try to take care of our end and try to bring momentum and help the team," said Gallagher, whose line also has tough winger Brandon Prust. "We're having a lot of fun as a team.
"It's early in the season, but it's going great so far."
The Canadiens built a 2-0 lead only to see the opportunistic Jets take the lead in the second period.
Cole tied the game at 15:08 of the second with assists from the rookies when he shoveled the puck in from the edge of the crease just after a power play ended.
Plekanec got the game-winner on a shot from the right circle that went in off defenseman Ron Hainsey's skate.
The Jets had two late power plays, including one when Prust was called for diving after being hit from behind by Antropov. They didn't convert.
"We had an opportunity to make it a game with two power plays in the third but we never got it in the zone with possession," Winnipeg coach Claude Noel said. "Then they come down and put one in off Hainsey's skate.
"It is what it is. It was 3-3 and we had a chance to win the game if we were patient, but we got the power plays and didn't capitalize."
Bourque scored on a shot from outside the right circle on a power play that Ondrej Pavelec waved at 8:04 into the game. Gallagher made it 2-0 with a shot off a rush that went in off the crossbar at 14:22.
"One of the nicer ones I'll score, I guess," said the left winger, who gets most of his goals jamming pucks from the crease.
Wheeler got one back as a bouncing shot got over Carey Price's arm on a power play at 16:10.
A Jets power play had just ended, but there was still heavy pressure in the Montreal zone when Antropov fired Bryan Little's rebound into an open side 3:32 into the second period.
Jokinen put Winnipeg ahead at 11:48 when he snapped a shot under the crossbar after a David Desharnais turnover in the neutral zone.
The Jets went 1 for 6 on the power play while Montreal's continued to click at 2 for 5.
The top players with the man advantage are point men Andrei Markov and Rafael Diaz, who had two assists each and share the team lead with seven points.
Markov missed most of the past two seasons with consecutive knee ligament surgeries. It wasn't long ago that he was the quarterback of one of the NHL's best power plays.
"I remember back when we played with Kovy (Alex Kovalev) and Marky (Markov) and I think we were first in the league two years in a row," Plekanec said. "I feel it's pretty much the same now.
"He has a huge impact on our team, not only on the power play."
Notes: The last time a team named the Winnipeg Jets won in Montreal was March 1985. ... P.K. Subban will join the Canadiens in Ottawa on Wednesday, but it is uncertain when he will play. He signed a two-year contract on Monday... Defenseman Tomas Kaberle returned to the Montreal lineup, bumping Yannick Weber.