Sutter, who has dual citizenship of both the United States and Canada, elected to play for Canada in international competition. Sutter represented Team Canada extensively during his junior career at the under-18 and under-20 levels. He competed in two IIHF World U18 Championships in 2006 and 2007, losing the bronze medal game in both tournaments.
Sutter is part of the venerable Sutter hockey family. He is the son of Brent Sutter, who coached him in junior with the Red Deer Rebels and Team Canada at the 2007 Super Series; Brent is a former head coach of the Calgary Flames. He has an older brother, Merrick who currently serves as the Rebels video coach and a younger sister, Brooke.
After a brief stint with Carolina's American Hockey League (AHL) affiliate, the Albany River Rats, at the end of his 2007–08 WHL season, Sutter debuted in the NHL with the Hurricanes in 2008–09. He scored his first NHL goal on October 23, 2008 against Marc-André Fleury of the Pittsburgh Penguins. The next game, on October 25, Sutter suffered a concussion after a collision with former Hurricane Doug Weight of the New York Islanders. Sutter had his head down as he was leaning forward for a loose puck in the neutral zone when Weight caught him with his shoulder. Although the hit was ruled as legal and Weight was not assessed any penalty, it re-sparked the debate in the NHL on head shots. Sutter returned to the line-up after missing eight games.
His cousin Brett was a teammate of his with the Rebels who was drafted two years ahead of him by the Calgary Flames, was the captain of the Charlotte Checkers with the Carolina Hurricanes organization, and currently plays in the Minnesota Wild organization. Another cousin, Brody, was formerly a forward for the Western Hockey League's Lethbridge Hurricanes and was drafted 193rd overall by Carolina in the 2011 draft. Brody is still with the organization playing the team's American Hockey League affiliate, the Charlotte Checkers.
On June 22, 2012, Sutter was traded to the Pittsburgh Penguins, along with Brian Dumoulin and Carolina's 2012 NHL Entry Draft first round pick (which the Penguins used to select Derrick Pouliot), in return for Jordan Staal.
On March 12, 2013, In a game against the Boston Bruins Sutter scored two goals, 3 minutes & 24 seconds apart, in the 3rd period that led to a 3-2 comeback victory for the Penguins.
On August 5, 2014, the Penguins announced they had re-signed Sutter to a two-year contract, worth $6.6 million.
On July 28, 2015, Sutter was traded by the Penguins along with a 2016 NHL Entry Draft 3rd-round pick to the Vancouver Canucks in exchange for Nick Bonino, Adam Clendening, and a 2nd-round pick in 2016. On August 4, 2015, the Canucks announced they had signed Sutter to a 5-year, $21.875 million deal. Sutter played 16 games in the 2015–16 season before it was revealed that he required Sports hernia surgery. He missed 33 games before returning to the Canucks lineup on January 26, 2016. However on February 9, his fourth game back since the surgery, he suffered a broken jaw in a game against the Colorado Avalanche; sidelining him for the remainder of the season. Sutter managed 5 goals and 4 assists (9 points) in 20 games for the Canucks in an injury-plagued 2015-16 season.
In August 2016, the Canucks announced that Sutter had switched from number 21 to 20 to let newcomer Loui Eriksson wear his old number 21. On January 4, 2017, Sutter was awarded his second career penalty shot where he scored against Mike Smith of the Arizona Coyotes.