Gallagher was born in Edmonton, Alberta, but grew up in Tsawwassen, British Columbia, after moving there with his family at the age of 12. As a youth, he played in the 2005 Quebec International Pee-Wee Hockey Tournament with a minor ice hockey team from Burnaby.

Playing in the South Delta Minor Hockey Association, Gallagher was selected in the ninth round of the 2007 WHL Bantam Draft by the Vancouver Giants. He joined the Giants for his WHL rookie season in 2008–09. Gallagher scored his first WHL goal against goaltender Kevin Armstrong on September 27, 2008, in a 7–3 win over the Prince George Cougars. After finishing the regular season with 10 goals and 31 points over 52 games, he added three points (one goal, two assists) in 16 post-season contests, as the Giants were eliminated in the semifinals by the Kelowna Rockets. The following season, Gallagher improved to 81 points over 72 games, ranking second among Giants players, behind Craig Cunningham, while his 41 goals were a team high. Gallagher’s efforts continued in the post-season, helping the Giants to the semifinals for his second consecutive year with the club. He registered 21 points (11 goals and 10 assists) in 16 games, second in team-scoring and fifth in the league, as Vancouver was eliminated by the Tri-City Americans.

During the 2010 playoffs, Gallagher was selected as the Male Youth Athlete of the Year in his hometown of Delta, British Columbia. In anticipation of the 2010 NHL Entry Draft, he was ranked 152nd among North American skaters eligible to be selected by the NHL Central Scouting Bureau. Gallagher was selected 147th overall by the Montreal Canadiens. Returning to junior following his first training camp with the Canadiens, he recorded WHL career highs in 2010–11. On February 28, 2011, he was named WHL Player of the Week after recording ten points (five goals and five assists) in three games. In March 2011, Gallagher was sidelined with a head injury, sustained after receiving a hit in a game against the Portland Winterhawks. Ranking first on the Giants and eighth among all WHL scorers with 91 points over 66 games, he was named to the WHL West First All-Star Team. His 44 goals were just four shy of Evander Kane’s single-season team record.

In his third WHL season, Gallagher was invited to the Canadian junior team’s World Junior selection camp in December 2010. Failing to make the roster, he was among the initial cuts. The following year, Gallagher made the team for the 2012 World Junior Championships, held in Alberta. In the last contest of the preliminary round, Gallagher was named Canada’s player of the game after recording a game-high seven shots on goal in a 3–2 win against the United States. In the semifinal, Gallagher recorded a goal and two assists on Canada’s last three goals, helping them get within one goal of erasing a 6–1 deficit against Russia. Canada went on to win their bronze medal game against Finland 4–0. Gallagher finished the tournament with six points (three goals and three assists), tying for fifth in team scoring, while leading Canada with 35 shots on goal. Gallagher was also a part of Team Canada in the 2016 World Championship. During the tournament, he scored 2 goals and 3 assists for a total of 5 points in ten games as Canada won gold.

Appearing in his second training camp with the Canadiens, Gallagher competed for a roster spot, remaining with the team until the final days leading up to the 2011–12 season. After being returned to junior, the Canadiens signed him to a three-year, entry-level contract on November 16, 2011. The following month, Gallagher temporarily left the Giants for the Canadian national junior team. In his first game back, on January 7, 2012, he recorded a seven-point night, which included a hat-trick, in an 8–4 win against the Portland Winterhawks. Three days later, he assumed the team captaincy after previous captain James Henry was traded away to the Moose Jaw Warriors. In the same week, during a game against the Tri-City Americans, Gallagher sustained an upper-body injury after colliding into the boards while pursuing a loose puck. Earlier in the shift, he had also received an open-ice hit. Gallagher returned to the line-up by late January and, the following month, he broke the Giants’ records for most goals and points all-time, both set by Adam Courchaine seven years prior. With a hat-trick against the Tri-City Americans on February 14, Gallagher surpassed Courchaine’s mark of 126 goals. Eleven days later, he recorded two goals in a 5–3 win against the Kamloops Blazers for his record-setting 274th point as a Giant. Near the end of the season, he was voted by WHL players, executives and broadcasters as the most valuable player to his team in the Tri-City Herald’s annual survey.

In his first professional season, Gallagher was directly assigned to the Canadiens’ American Hockey League (AHL) affiliate, the Hamilton Bulldogs, due to the 2012–13 NHL lockout. Upon a resolution, Gallagher was recalled to the Canadiens’ training camp for the shortened 2012–13 season. He scored his first NHL goal on January 27, 2013, against Martin Brodeur in a 4–3 win over the New Jersey Devils. After the trade of Michael Ryder back to the Canadiens, Gallagher switched from jersey number 73 to 11 to accommodate the veteran Ryder, who has worn 73 throughout his NHL career. It was later revealed in 2019, that Ryder offered Gallagher a Rolex in return of the number 73, Ryder never filled his promise. Gallagher ended the season with a total of 15 goals and 13 assists in 44 games and established himself as a key member of Montreal’s offence, often playing on the Canadiens’ first line. On May 6, 2013, Gallagher was also nominated for the Calder Memorial Trophy, awarded to the NHL’s Rookie of the Year, and placed second in voting.

On November 29, 2014, the Canadiens came to terms with Gallagher on a six-year contract extension.

In January 2014, McDonald’s introduced the “Gallagher burger” in its Quebec restaurants. Gallagher again created a signature burger for McDonalds before the 2016 NHL season, along with teammate Alex Galchenyuk (who created one of his own).

On September 18, 2015 it was announced that Gallagher would become one of the alternate captains of the Montreal Canadiens along with P. K. Subban, Tomáš Plekanec, and Andrei Markov. During the 2015–16 season, Gallagher broke two fingers after blocking a shot from Johnny Boychuk and would require surgery to fix them. He returned to the Canadiens lineup for the 2016 NHL Winter Classic.

For the 2018–19 season, Gallagher was once again named an alternate captain for the Canadiens along with Paul Byron. On February 21, 2019, Gallagher recorded his first career hat trick in a 5-1 win against the Philadelphia Flyers. October 30th 2019 he played in his 500th career game in the NHL and scored a goal that night to beat the Coyotes in Phoenix, Arizona. Brendan was awarded the Molson cup by the Montreal Canadiens for the month of October 2019.

On October 14, 2020, Gallagher signed a six-year, $39 million contract extension with the Canadiens.