The 12th seed was dominant from the start, taking control to win the opening set in 29 minutes despite several strong serves from his opponent.
Anderson, who needed treatment on his elbow at the end of the first set, fell an early break behind in the second as Djokovic continued to dominate.
The South African went up a gear in the third, holding his first four service games, although Djokovic proceeded to save five set points.
Djokovic was then peerless in the tie-break, getting the job done to seal his 13th Grand Slam title.
The 31-year-old's first major for two years moves him moves into outright fourth place on the all-time list of Grand Slam men's singles titles, moving clear of Roy Emerson.