The corporation announced that it had dropped 66-year-old choreographer Phillips in favour of popstar Alesha Dixon, 30, last week following weeks of speculation about her future.
Songwriter Stilgoe, believed to be a close friend of Phillips, has now branded the decision "awful and ageist" in an interview with the Daily Mail.
He added: "Arlene knows stuff, is brilliant. I can't believe any viewers think she ought to go. It's an older person being replaced by a younger person. It's just a pity."
Stilgoe, also 66, reportedly stayed in touch with Phillips after they worked together on musical Starlight Express in the 1980s.
Former Strictly contestants Fiona Phillips and Rachel Stevens have both spoken out in defence of the ex-panel member in recent days. Head judge Len Goodman has also expressed scepticism over the show's revamp.
Meanwhile, reports suggest that the programme's professional dancers believe the decision to hire Dixon is "an absolute farce".