Jonathan Ross was cleared to keep his £6 million-a-year BBC job despite being criticised by the corporation's governing body over two offensive broadcasts.
Repeated failures of editorial judgment allowed swearing and other obscene material to be aired in a series of TV and radio programmes, the BBC Trust said.
The Trust said the worst case was an episode of Radio 2's Russell Brand Show which included obscene messages left on Sachs's answerphone by Ross, 48, and Brand, 33.
The two presenters said Brand had slept with the 78-year-old Fawlty Towers actor's granddaughter, Georgina Baillie, 23, and joked that Sachs might kill himself.
It also condemned an edition of BBC1's flagship chat show, Friday Night with Jonathan Ross, in which the host said he would "f***" actress Gwyneth Paltrow if his wife permitted.
In the furore that followed the airing of the Sachs calls on October 18, Brand resigned and Ross was suspended without pay for 12 weeks.
Radio 2's controller, Lesley Douglas, and head of compliance, David Barber, also quit their jobs after clearing the programme for broadcast despite knowing about its contents. The BBC is still facing an investigation by regulator Ofcom, which has the power to impose a fine of up to £250,000.
Delivering its report into the row at Broadcasting House in central London, the BBC Trust condemned both the recording and the airing of the material. But it said it would not be seeking further sanctions against Ross, meaning he will be able to return to his presenting job early next year.
Radio 2 has said he will be back on air on January 24 but BBC1 has not confirmed when he will resume his TV show.
BBC Trust chairman Sir Michael Lyons said: "We are very clear that the Director-General has taken the right action with respect to Jonathan Ross."
