On July 1, 2016, Connolly signed to a one-year contract as a free agent with the Washington Capitals. On July 1, 2017, he signed a two-year contract extension with the Capitals. In his first season with Washington, Connolly recorded a career-high 15 goals while appearing in 68 games. The following season, in 2017–18, he equaled that career best in goals while also recording a career-high 27 points. He appeared in all 24 playoff games for the Capitals in 2018 while scoring six goals and three assists, including the primary assist on Lars Eller's Stanley Cup-winning goal with 7:37 left in the third period of Game 5 of the 2018 Stanley Cup Finals. On July 1, 2019, having left the Capitals as a free agent after three seasons, Connolly was signed to a four-year, $14 million contract by the Florida Panthers. 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 eight points (three goals and five assists) over six 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.