Both players held serve throughout the first set, but Anderson managed to break Brands at 7-5 to take an early one-set lead.
However, Brands fought back straight away, breaking Anderson in the first game of the second.
The South African was unable to break back and saw Brands serve out for the set, 6-4.
The third set was an altogether more one-sided affair, as Anderson got the better of Brand's serve to claim the set 6-2, hitting strong serves, including one at 134mph.
Anderson continued his good play in the fourth, again breaking Brand to move into a commanding position at 4-2.
After that the South African served for the match and duly won after a two-shot rally.