In reply to SA's paltry first-innings total of 128, WA were 6-257 at tea on day two, with D'Arcy Short on 23 and Joel Paris on three.
Green (61 off 94 balls) and Cameron Bancroft (46 off 134 balls) combined for a 101-run stand on a green deck to guide WA out of early trouble, before Hilton Cartwright (73) and Josh Philippe (32) powered home the advantage.
With four first-innings wickets still in hand, WA have a lead of 129.
Bancroft was the first to fall on Thursday when he was caught behind off speedster David Grant (3-62).
Green looked ominous during a knock that featured seven fours and a six.
But he fell shortly after Bancroft's dismissal when he skied an attempted pull shot off Grant.
Green started the domestic season with a classy 106 against South Australia.
But his next five knocks in Shield and one-day cricket have returned scores of two, five, seven, 13 and nine in what marked a rare form slump for the 22-year-old.
His half-century on Thursday was welcome news for both WA and Australia's national selectors, who view Green as a vital part of the Ashes puzzle.
Cartwright struck eight fours and two sixes during his 154-ball knock before becoming Grant's third victim, trapped LBW.
Redbacks pace bowler Brendan Doggett was forced off the field during his fifth over on Thursday with what appeared to be a side strain.