Shadow leader of the house Valerie Vaz joked that cries of "Don't panic! Don't panic!" were being heard around 10 Downing Street as the government tries to organise its policies on Brexit.
Commons Leader David Lidington speculated that his Labour opposition "don't like it up 'em'".
Mr Perry, who also created sitcom Hi-de-Hi, died on Sunday aged 93.
During the business statement, Ms Vaz said she grew up watching Dad's Army, adding: "It struck me that you could hear those catchphrases ringing around Number 10, so we hear the cry of 'Don't panic! Don't panic!'
"Or as the Prime Minister slaps down her recalcitrant and wayward colleagues you can hear her muttering 'stupid boys'.
"When you ask their position on Brexit, the infamous 'don't tell 'em Pike!'."
Mr Perry worked with the late producer David Croft and the pair were also responsible for It Ain't Half Hot Mum, Hi-de-Hi and You Rang M'Lord?
Commons Leader David Lidington also paid tribute to Mr Perry, who often wrote the theme tunes to his shows.
He said: "I thought it was a wonderful gesture when outside Buckingham Palace earlier this week the military band in the changing of the guards ceremony played the theme tune to Dad's Army as a tribute.
"But I do think when I look across at the Opposition, in particular when I look at their faces during Prime Minister's Questions, the phrase that comes into my mind is 'they don't like it up 'em'."
The SNP's Commons leader Pete Wishart also chipped in, saying: "I would hate to say that we are all doomed but perhaps we are under this particular Government."
TV references also came up in the Commons on Wednesday at Prime Minister's Questions when Jeremy Corbyn likened Theresa May's plan for the UK leaving the UK as one of Baldrick's "cunning plans" from the Blackadder series.

Lee Gregory could have put the Lions ahead, but shot over from close range.
Visting goalkeeper Jordan Archer made a fine save to deny Elliot Lee before Gregory and Chris Taylor both missed chances at the other end.
Taylor almost sealed three points for Millwall late on but his defected effort hit the post, meaning the Londoners drop to sixth in the table.
They are four points ahead of Barnsley, who have a game in hand, while Colchester remain in 23rd place, still seven points adrift of safety.
Millwall boss Neil Harris told BBC Radio London:
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"I don't think we were really quite as clinical as we have been.
"I thought we were very lacklustre in the first half and never really imposed our tempo on the game.
"I think we can focus on the fact that it's a really good clean sheet, it's another point gained.
"Having said that, I'm obviously disappointed. I thought we should have come here and won today."
MPs have paid tribute to Dad's Army creator Jimmy Perry, using his show's famous catchphrases in the Commons.
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League One strugglers Colchester United had to settle for a point against play-off chasing Millwall.