Gibson was selected to represent the United States in the 2013 World Junior Ice Hockey Championships. He was the team's primary goaltender throughout the championship, registering a 95.54% save percentage and a 1.36 goals against average in seven games for the gold medal winners. His save percentage led all goaltenders in the tournament and he was named as the tournament's best goalkeeper. He was also named to the tournament All Star team as well as the tournament's Most Valuable Player. Gibson also won a bronze medal at the 2013 IIHF World Championships, posting a 1.56 goals-against average and .951 save percentage.
On April 7, 2014, aged 20 years and 297 days, Gibson made his first NHL start for the Anaheim Ducks following an injury to goaltender Frederik Andersen. Making 18 saves for a shutout, Gibson earned his first NHL win, a 3–0 victory over the Vancouver Canucks. In doing so Gibson became the youngest NHL goalie to record a shutout in his NHL debut since Buffalo goalie Daren Puppa (20 years, 223 days) in the 1985–86 season.