Stephen Doughty, Jonathan Reynolds and Kevan Jones all resigned after shadow Europe minister Pat McFadden was sacked by leader Jeremy Corbyn over "disloyalty". In an interview with the BBC’s Daily Politics programme, Austin labelled the reshuffle as "a shambles from start to finish".
In another interview with the BBC he later said the leadership "should reflect on they way this has been done".
Austin went on to say: "I would never have appointed someone with his [Milne’s] views and background to a senior position in the Labour in the first place. But I think his behaviour over the last few weeks has been a disgrace."
Kevin Slocombe, who is Corbyn's head of media, was asked to comment on Austin's allegation, and told PRWeek via text message: "It's untrue."
Earlier in the afternoon, Prime Minister David Cameron had publicly mocked the drawn-out reshuffle during a tense PMQs.
Milne, who was working as a Guardian columnist and associate editor, joined the Labour Party as executive director of strategy and comms in October 2014. His role includes handling comms for the Labour Party itself, and the office of Corbyn.