The linemates put up two points each, with 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 night.
"There's always going to be pressure on me, but you don't want to take your foot off the pedal," said the stocky, five-foot-eight 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 tie-breaking power play goal 5:31 into the third period was the game-winner for Montreal (4-1-0).
Rene Bourque and Erik Cole each had their first of the season for the Canadiens, while Olli Jokinen had his first as a Jet. Nik Antropov and Blake Wheeler also scored for Winnipeg (3-2-1), which ended a three-game winning run.
First-year general manager Marc Bergevin ended debate over whether Galchenyuk should be sent back to the junior Sarnia Sting and Gallagher to the AHL Hamilton Bulldogs 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 honours and prolonged applause from the 21,273 Bell Centre fans. Galchenyuk was named 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 unit 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.
Cole tied the game at 15:08 with assists from the rookies when he shovelled the puck in from the edge of the crease just after a power play ended.
Plekanec got the game-winner on a blast from the right circle that went in off defenceman 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," said Winnipeg coach Claude Noel. "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 blast from outside the right circle on a power play that Ondrej Pavelec waved at 8:04 into the game, and 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 frame.
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 guns 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 last two seasons with consecutive knee ligament surgeries. It wasn't long ago that he was quarterback of one of the NHL's best power play units.
"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," said Plekanec. "I feel it's pretty much the same now.
"He has a huge impact on our team, not only on the power play."
Backup Peter Budaj is set to make his first start of the season in goal as the Canadiens face the Senators in Ottawa on Wednesday night.
The Jets started a three-game road trip that takes them to Florida on Thursday and Tampa on Friday.
Notes: The last time a team named 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. Likely on the weekend. ... Defenceman Tomas Kaberle returned to the Montreal lineup, bumping Yannick Weber. ... Rookie Mark Scheifele was among Winnipeg's scratches along with Anthony Peluso and Zach Redmond.