Six days ago City beat Brighton 4-1 to retain the Premier League title. In February they claimed the League Cup after a penalty shoot-out win over Chelsea.
"It was an incredible final for us and we have finished an incredible year," said City manager Pep Guardiola. "To all the people at the club a big congratulations, especially the players because they are the reason why we have won these titles."
Sterling is the first player since Blackpool's Stan Mortensen in 1953 to score three times in an FA Cup final. David Silva, Kevin de Bruyne and Gabriel Jesus added City's other goals.
It was the biggest FA Cup final win since Bury beat Derby 6-0 in 1903.