Philander took four for 29 as the West Indies were bowled out for 201 and forced to follow on, 351 runs behind South Africa's first innings total of 552 for five declared.
Philander struck again early in the second innings when he dismissed Devon Smith for the second time in the day.
West Indies were 76 for two at the close, still needing 275 runs to avoid an innings defeat.
There was a setback for South Africa, however, as strike bowler Dale Steyn limped off the field after bowling only five balls in the follow on innings.
Philander, who had taken only four wickets at high cost in his previous four Test matches, did not take a wicket in his opening spell on Friday but transformed the match when he came back for a second spell.
After a good start by opening batsmen Kraigg Brathwaite and Smith, the West Indian batting disintegrated in dispiriting fashion.
Brathwaite and Smith saw off South Africa's new ball attack, putting on 72 for the first wicket before Philander dismissed both openers in successive overs.
Smith went on to make 35 before being caught behind off Philander's fourth ball of his second spell.
He was given not out by umpire Billy Bowden but Philander immediately asked for a review which showed a faint bottom edge as Smith played a cramped pull shot.
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