The team announced the moves Monday.
MacLellan, who also gets the title of senior vice-president, replaces George McPhee, whose contract was not renewed after 17 seasons with the Capitals.
MacLellan has been with Washington for 13 seasons, seven as assistant GM. He played for five teams during a 10-year NHL playing career.
Trotz was the Predators' coach for 15 seasons before being dropped from the job in April. In Washington, he takes over for Adam Oates, who was fired with a season left on his three-year deal.
This season, the Capitals missed the playoffs for the first time in seven years.