Lord Mandelson had written to Epstein in 2008 while he faced charges, telling him: “Your friends stay with you and love you.”
Sir Keir Starmer had defended Mandelson in the Commons just 24 hours earlier.
The decision to sack him prompted Conservative MPs to accuse the Prime Minister of a “total fiasco” with only days to go until the state visit of US President Donald Trump.
Shadow Housing Secretary James Cleverly said: “About a week out from the state visit of President Trump we have no ambassador and a Foreign Secretary that has been in the job for only a few days. What a total fiasco!!! And this is all Starmer’s fault. I can see him being gone by Christmas.”
Mr Polanski, who has branded himself an “eco-populist” and launched a podcast which surged to fourth in the charts, said the scandal proved politics was rigged to reward failure.
He added: “Peter Mandelson will be fine. He’ll go have a job now with a high paid lobbying firm or he will fail upwards.
“And we see this throughout our society constantly. I think it’s about a broken system of politics and a broken system of lobbying that seems to allow some of the people with the worst judgments, the worst values and the worst connection to the problems of everyday people in our country to constantly be promoted and constantly successful.”
Labour was contacted for comment.