Sautner was selected by the Edmonton Oil Kings in the eighth round (162nd overall) of the 2009 WHL Bantam Draft. During the 2013–14 season, Sautner's plus/minus record was the best in the WHL.
Sautner went to the Minnesota Wild prospect camp prior to the 2012-13 season. He attended the Arizona Coyotes prospect camp in the 2014 pre-season. During his fourth season with the Oil Kings, despite being undrafted, Sautner was signed to a three-year entry level contract with the Vancouver Canucks on March 14, 2015. On March 24, 2018, the Canucks recalled Sautner on an emergency basis. He made his NHL debut the following night, a 4–1 Canucks victory over the Dallas Stars. Three games later, on March 31 against the Columbus Blue Jackets, Sautner earned his first NHL point, an assist on a Darren Archibald goal.
Sautner began the 2018–19 season with the Utica Comets in the AHL after clearing waivers. On February 19, he was recalled to the NHL for the first time that season.