The show saw the pair make obscene comments about Sachs' 23-year-old granddaughter, Georgina Baillie.
The Trust has demanded an on-air apology and called for editorial standards in the BBC to be tightened, with particular attention to "high risk" programmes.
In a statement issued after their meeting with Thompson, the Trust said it was "dismayed" that the broadcasts "fell so far short of audience expectations".
It said there had been a "deplorable intrusion" into the privacy of Mr Sachs and his granddaughter."
They issued an unreserved apology to licence fee payers, as well as to the Fawlty Towers actor and his family.
Douglas, who took up her post as controller in 2003, admitted in her resignation letter that it had been a "painful week" for Radio 2.
She said: "I am proud of what we have achieved together over the years and I shall miss you all."
Russell Brand resigned from his Radio 2 show
"The events of the last two weeks happened on my watch. I believe it is right that I take responsibility for what has happened," she wrote.
In his response, Thompson said the decision had been accepted with "real sadness".
Speaking on the BBC News Channel, he said: "Given the scale of the BBC output it is inevitable from time to time programmes will encounter difficulties, people will make editorial misjudgements."
"The ultimate editorial responsibility for BBC programmes lies with producers and editorial managers."
Alex Lester, Radio 2's weekday morning presenter, will replace Brand as the station's new Saturday late night host, following his resignation on Wednesday.
BBC One's Friday Night with Jonathan Ross has been cancelled, and the 1994 film Speed will be shown in its place.
Ross's Saturday morning radio show has also been pulled from Radio 2's schedules. His regular slot will be occupied by Richard Allinson.
A Sunday morning repeat of Ross's Film 2008 is being substituted with an edition of natural history programme Wild. However, a decision about what will replace Wednesday's show has not yet been taken.
Tim Davie, director of BBC Audio & Music, said: "What is important now is that we remain focussed on creating great programmes for our listeners which remains, as always, the number one priority of the group."
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