In a statement, Ofcom said the scale of the fine "reflects the extraordinary nature and seriousness of the BBC's failures and the resulting breaches of the [Broadcasting] Code".
The two men made the calls in Brand's pre-recorded Radio 2 show and it was approved for broadcast despite Sachs saying he was unhappy with the stunt.
The incident, which caused a public outcry last October, became known as Sachsgate and resulted in the resignation of Brand, Radio 2 controller Lesley Douglas and the station's head of specialist music and compliance David Barber. The BBC also punished Ross with a three-month unpaid suspension.
The fine covers breaches of two rules in Ofcom's Broadcasting Code. The first, of £80,000, was imposed because there was deemed to be an unwarranted infringement of privacy.
The second, of £70,000, was as a result of a failure to apply "generally accepted standards" to the content of the radio show "so as to provide adequate protection for members of the public".
Ofcom also deemed that the show included material "which may cause offence" but is "not justified by the context".
The fine applies to two editions of Radio 2's The Russell Brand Show, the original on October 18 and another on October 25 when Brand and Ross 'sang' apologies to Sachs.
The offensive material for which the presenters came under fire related to claims left on Sachs's answerphone that Brand had slept with the actor's granddaughter, aspiring actress and burlesque dancer Georgina Baillie.
On one of the messages Ross shouted "he f***ed your granddaughter". Ofcom described this incident as "particularly gratuitous" and said it was compounded by the "prolonged nature" of the presenters' discussions about Baillie.
Public complaints about the incident, with the Daily Mail leading a backlash by some national papers, rose to more than 44,000 - 1,939 to Ofcom and 42,851 to the BBC.
In its report Ofcom outlined a number of "key issues" that had arisen from the incident which included "a failure of editorial judgment in relation to both the recording and broadcasting of offensive and intrusive material", a conflict of interest because a BBC producer was loaned out to the independent production company to make the show, the compounding of the problem by the broadcast of a sung apology a week later, and "failures of compliance systems".
Ofcom also received a complaint about the October 21 episode of Radio 1's 'Chris Moyles Show' on which Brand was a guest and the incident was discussed, together with his association with Baillie.
The BBC Trust has issued a reponse to today's fine, saying it regrets the serious breaches of the BBC that "have led to a financial penalty... and the loss of licence fee payers' money as a result".
The Trust's Editorial Standards Committee concluded that the content under discussion was "grossly offensive" and represented "an abuse of the privilege given to the BBC to broadcast to its audiences". It identified a failure to assert editorial control by Radio 2, a failure to follow the compliance systems in place and a failure in editorial judgement.
A spokesman for the BBC said the corporation accepted Ofcom's findings, saying: "This material should never have been broadcast and we apologised unreservedly for that.
"We note that Ofcom has found that senior management acted swiftly to mitigate the offence and damage caused by the breaches of the code."
He went on to confirm there had been a "comprehensive review" of compliance across Audio and Music since the incident and that "a new policy ensuring presenters and their agents cannot be the executive producer responsible for compliance on their own shows" was now in place.
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