Mr Burnham said the BBC Trust has taken a "range of actions"
Russell Brand and Jonathan Ross's lewd phone calls to Andrew Sachs were a "serious lapse" of standards, Culture Secretary Andy Burnham has said.
He added that BBC management had been "too slow" in recognising the seriousness of the situation.
After the calls to actor Andrew Sachs were broadcast, Ross was suspended for three months from the BBC while Brand quit his Radio 2 show.
Mr Burnham was speaking during Commons question time.
The issue was raised by shadow culture secretary Jeremy Hunt.
'Range of actions'
Radio 2 controller Lesley Douglas and the station's head of specialist music and compliance, David Barber, also resigned.
Mr Hunt had told ministers that viewers were confused over who to complain to about the incident.
He said: "It's difficult for viewers to know who to complain to if they are unhappy with BBC programmes that are funded by their licence fee - whether it's BBC management, BBC Trust or Ofcom."
He added: "It is absolutely essential that viewers are represented by an independent body that champions the needs of licence fee payers and not by an organisation that defends the BBC as an institution."
Mr Burnham replied that the BBC Trust - which is independent of the BBC itself - had taken a "range of actions", including a review of the BBC's editorial guidelines, to ensure there was no repeat of the prank calls row.
With regard to complaints, he said the BBC Trust was equipped to deal with issues raised by licence fee payers, while Ofcom looked at the wider view of broadcasting to ensure standards "across the board" were upheld.
The offending broadcast, which featured Ross as a guest on Brand's Saturday night radio show, attracted 42,000 complaints.
On Saturday, Radio 2 broadcast two apologies. The first went out at 1003 GMT, when Ross's show would have gone out, and a fuller version was broadcast again at 2113 GMT during Brand's former slot.