According to the corporation, Downing Street requested Campbell be replaced with a shadow minister on the show, but the programme's executive editor Gavin Allen refused as a point of "fundamental principle".
Allen said that the situation resulted in no representative of the coalition government featuring on the country's "most-watched political programme in Queen's Speech week - one of the most important moments in the Parliamentary calendar".
The panel instead consisted of former Lib Dem MP Susan Kramer, author Sir Max Hastings and former newspaper editor Piers Morgan. However, Tory backbencher John Redwood also contributed to the discussion.
Introducing the programme, Question Time host David Dimbleby said that he would have "expected" to see a government minister on the panel, particularly in the week that the coalition unveiled its legislative agenda.
He also explained that No 10 had said that a minister was "available" - who was believed to be Chief Treasury Secretary David Laws - but they would only appear if Campbell was replaced.
Dimbleby added that it was up to the Question Time producers to "decide who should be on the programme not Downing Street".
Speaking about the situation, Campbell said that he would have "expected" the programme to feature a government minister in the week of the Queen's Speech, "regardless of who else is on the panel".
In response, a Downing Street spokesman said: "In the week of the Queen's Speech the BBC booked Alastair Campbell in the place of an opposition front bencher to appear on Question Time - which we questioned.
"Before a final decision was made on who might appear on behalf of the government the BBC directly booked John Redwood MP."
Writing in a BBC blog, Allen said that the government objected to Campbell not being an "elected Labour representative or a front-bencher".
However, he dismissed the objection and instead backed Campbell as "one of the most senior and influential figures in the Labour movement".
"It is a fundamental principle of our independence that politicians cannot dictate who sits on the panel," said Allen.
"It is for Question Time, not for political parties, to make judgements about impartiality and to determine who is invited to appear in the interests of the audience.
"Parties are free of course to accept or reject those invitations, but they do not have a right of veto over other panellists. Licence fee payers rightly insist that the BBC must be free from political interference."