The reigning champion beat the South African 20th seed 6-4 6-3 7-6 (8/6), continuing his flawless run at the All England Club this year.
Murray has not dropped a set yet and will be back on court on Wednesday for his last eight tie where he is likely to next meet Queen's champion Grigor Dimitrov.
The Centre Court roof, which had been shut for most of the preceding match, opened before Murray and Anderson walked onto Centre Court.
That would no doubt have pleased the Scot, who prefers playing outdoors, particularly when his opponent is a big hitter like Anderson.
Murray's great supporter Sir Alex Ferguson watched another victory from the Royal Box.