Internationally, Connolly has represented Canada at the 2009 IIHF World U18 Championships, 2009 Ivan Hlinka Memorial Tournament, 2011 IIHF U20 Championships and 2012 IIHF U20 Championships.
Raised in Prince George, British Columbia, Connolly was selected by his hometown Prince George Cougars as their first selection, tenth overall, in the 2007 WHL Bantam Draft. As a 15-year-old, he played major midget for the Caribou Cougars in Prince George, while also appearing in four WHL games. He joined the WHL's Cougars full-time in 2008–09, scoring 30 goals and 30 assists in 65 games. In doing so, he became the first 16-year-old to score 30 goals in the WHL since Patrick Marleau performed the feat in 1995–96. He was named the winner of the Jim Piggott Memorial Trophy as WHL rookie of the year. The recognition made him one of three nominees for the Canadian Hockey League (CHL) Rookie of the Year; he beat out the Quebec Major Junior Hockey League (QMJHL)'s Dmitri Kulikov and the Ontario Hockey League (OHL)'s Evgeny Grachyov for the national distinction.
Conolly's second full WHL season was marred with hip problems. He played in just 12 of the Cougars' first 46 games and was sidelined from December 2009 before returning for the last four contests of the regular season. He was named to play in the 2010 CHL Top Prospects Game, but did not participate. He recorded 19 points over 16 games.
Connolly was highly ranked for the 2010 NHL Entry Draft. He was listed as the sixth-best player amongst North American skaters for the draft in the International Scouting Services' midterm rankings and finished the season 13th among all skaters. The NHL Central Scouting Bureau ranked him fourth overall amongst North American skaters in their midterm rankings, before moving him up a spot in their final rankings. The Hockey News listed him at fourth overall in their list end-of-year list. Scouts described Connolly as a natural leader with good on-ice awareness who plays in the style of a power forward. Connolly compared himself to Columbus Blue Jackets power forward Rick Nash. Connolly was selected sixth overall in the draft by the Tampa Bay Lightning.
Attending his first NHL training camp in September 2010, Connolly was returned to the Cougars to continue developing at the major junior level for the 2010–11 season. Whilst playing in his fourth WHL season, he was named captain of the Cougars and recorded 73 points over 59 games, leading his team in scoring.
Connolly made the Lightning's NHL roster in October 2011 out of training camp. He then scored his first career NHL goal on November 1 against Cam Ward of the Carolina Hurricanes. His first professional season, however, was interrupted when he was loaned to Team Canada for a second-straight appearance at the IIHF World U20 Championships.
On January 10, 2012, Prince George traded Connolly's WHL rights to the Tri-City Americans in exchange for a bantam draft pick and two conditional draft picks should Connolly return to the WHL; the condition was never met. Connolly finished his rookie NHL season with 68 games played and 15 points scored (four goals and 11 assists).
On September 14, 2012, the Tampa Bay Lightning assigned Connolly, as well as 17 others, to the American Hockey League (AHL)'s Syracuse Crunch during the 2012–13 NHL lockout. On April 4, 2013, Tampa recalled Connolly from the Crunch. In the AHL, he appeared in 67 games, registering 27 goals and 57 points to go along with a +15 plus-minus rating and 51 penalty minutes. He ranked second on the Syracuse roster in both goals and points, and third for assists with 30. After a five-game stint in which he scored one goal, on April 12, the Lightning reassigned Connolly back to the AHL. Connolly would then remain with Syracuse, where he helped the Crunch reach the Calder Cup Final against the Grand Rapids Griffins, though the team would be defeated in the six-game series.
In preparation for the upcoming 2013–14 season, Connolly had a strong training camp with the Lightning, scoring four goals during the pre-season. However, on September 29, Tampa Bay assigned Connolly to Syracuse as the team finalized its NHL roster. Connolly's demotion was in large part due to the impressive play of rookies Tyler Johnson, Ondřej Palát and Richard Pánik, who as a line had posted five goals and ten points in four pre-season games; the line was expected to serve as Tampa's third line, thereby relegating Connolly to limited fourth line minutes. Explaining the demotion, Lightning General Manager Steve Yzerman stated, "I don't want Brett playing on the fourth line. I want him playing a lot of minutes, he's got to go [to Syracuse] and be our top guy, be a leader and carry the team, if need be. That's another step for him. There's no plan to leave him there for any period of time."
On October 17, however, the Lightning recalled Connolly from Syracuse, appearing in one game with the team before again being reassigned on October 20. On October 31, the Tampa recalled Connolly, where he played in six games, scoring game-winning goal against the St. Louis Blues on November 2 before being demoted again on November 22. Connolly was named to the 2014 AHL All-Stars roster. He finished his NHL season with 11 games played and one goal scored while also playing in 66 AHL games, scoring 21 goals and 57 points, leading the Crunch in the latter statistic.
On July 10, 2014, the Lightning re-signed Connolly to a one-year, two-way contract. On October 24, Connolly played in the first period against the Winnipeg Jets, but he would miss the rest of the game with a lower body injury. After being out for nearly a month, Connolly returned to the lineup against the Toronto Maple Leafs on November 20. On December 6, Connolly played in his 100th career NHL game in 3–1 Lightning loss to the Columbus Blue Jackets.
On March 2, 2015, at the NHL trade deadline, Connolly was traded to the Boston Bruins in exchange for two second-round draft picks in 2015 and 2016. On March 4, he fractured his hand in his second skate with the Bruins. On April 3, Connolly made his debut with the Bruins, recording two assists in Boston's 3–2 win over the Detroit Red Wings.
On July 6, 2015, Connolly agreed to re-sign with the Bruins on a one-year deal as a restricted free agent.
Unable to live up to his draft expectations with the Lightning and the Bruins, Connolly as a restricted free agent was not tendered a contract by Boston. On July 1, 2016, Connolly signed to a one-year deal as a free agent with the Washington Capitals. On July 1, 2017, Connolly signed a two-year extension with the Capitals.
Connolly made his international debut with Hockey Canada, representing Team Pacific at the 2009 World U-17 Hockey Challenge. He helped the team to the gold medal game, where they lost 5–1 to Team Ontario. Connolly recorded 8 points (3 goals and 5 assists) over 6 games. Several months later, he joined Team Canada for the 2009 IIHF World U18 Championships. He scored six points in six games as Canada placed fourth. Connolly remained with the national under-18 team for the Ivan Hlinka Memorial Tournament in August 2009. Connolly recorded an assist in the gold medal game, helping Canada to a 9–2 win over Russia. During the tournament, Connolly hurt his hip, a nagging injury that sidelined him long-term throughout the following season.
Connolly returned to the under-18 team the following year for the 2010 IIHF World U18 Championships in Minsk and Bobruisk, Belarus. Connolly recorded one goal over four games, as Canada finished in fourth place for the second consecutive year.
Moving on to the under-20 level, Connolly joined Canada's junior team for the 2011 World Junior Championships in Buffalo, New York. He recorded three assists over seven games, as Canada won silver. They lost in the gold medal game to Russia. Playing in his NHL rookie season the following year, the Tampa Bay Lightning loaned Connolly to the national junior team for the 2012 World Junior Championships.