Selected by the Blue Jackets in the fourth round (No. 94) of the 2009 NHL Draft, Savard set a Quebec Major Junior Hockey League record in 2009-10 for assists by a defenseman with 64 for Moncton.

That record, and Savard's plus-32 rating, helped him earn three league awards that season: the Kevin Lowe Trophy, given to the best defensive defenseman; the Emile Bouchard Trophy, given to the defenseman of the year; and the Canadian Hockey League's Defenseman of the Year.

Savard made his NHL debut against the Nashville Predators on Oct. 7, 2011 and scored his first goal against the Minnesota Wild on Feb. 7, 2012.

He set NHL career highs in goals (11), assists (25) and points in 2014-15, and tied for the lead among Blue Jackets defensemen with 25 points (four goals, 21 assists) in 2015-16.

Savard played in all 82 regular-season games in 2018-19 and scored 24 points (eight goals, 16 assists), then helped the Blue Jackets win a Stanley Cup Playoff series for the first time, a four-game sweep of the Tampa Bay Lightning in the Eastern Conference First Round. Savard had one goal and two assists in 10 playoff games.

He was traded to the Lightning in a three-team trade with the Detroit Red Wings on April 10, 2021. Savard did not score a point in 14 regular-season games but scored five points, all assists, in 20 playoff games to help the Lightning win the Stanley Cup for the second consecutive season.

An unrestricted free agent, Savard signed a four-year contract with the Montreal Canadiens on July 28, 2021. He played 18:58 in his 800th NHL game, a 4-1 loss to the Los Angeles Kings on Oct. 17, 2024, and retired from the League following Montreal's 4-1 loss at the Washington Capitals in Game 5 of the first round April 30, 2025.