Earlier, wicketkeeper-batsman Jonny Bairstow, who became the 15th England batsman to be out one short of his century, added 50 runs for the final wicket in alliance with Anderson, helped England post 362 in their first innings. During the interval, England made a clear decision to break the shackles, and although Jonny Bairstow was caught at deep backward square-leg to give Olivier a third victim, Moeen was able to continue the assault thanks to Elgar's second drop - this time off Maharaj.
England remain strong favourites to win the fourth cricket test against South Africa after Moeen Ali's late strokeplay arrested their second innings slide.
Anderson had success with the third ball of the South African innings when he removed Dean Elgar lbw for a duck with a trademark swinging delivery that beat the opener and rapped into his front pad.
However, the England and Wales Cricket Board (ECB) has said that they have not been contacted by Cricket South Africa and won't entertain any offers until the ongoing four-Test series against the Proteas concludes on Tuesday.
"I think it was my best (innings for England) in terms of trying to take momentum and putting pressure back on them", Moeen said. "But I'm thrilled by the honour and it's nice to get some wickets", the Guardian quoted Anderson as saying.
Rabada removed Toby Roland-Jones (11) and Moeen lofted Maharaj over midwicket for another maximum to further underline England's position of strength before the rain that had threatened since lunch finally arrived.
"We have had no approach from Cricket South Africa", the European Central Bank statement said.
It would leave England searching to appoint a replacement before the start of this winter's Ashes in November, with former Australia bowler and Yorkshire coach Jason Gillespie a possible interim appointment for that series.
Broad had earlier claimed the final wicket of South Africa's first innings.
Olivier's fall meant James Anderson was denied a maiden five-wicket Test innings haul at his Lancashire home ground.