He said the Prime Minister will "try anything" to avoid taking the blame for the unfair treatment of some British people who arrived here from Commonwealth countries.
The Prime Minister attacked Mr Byrne, Labour MP for Hodge Hill, as she answered questions in the House of Commons.
She pointed out that Mr Byrne had announced plans to create a "hostile environment" for illegal immigrants, when he was the immigration minister in a Labour government in 2007.
The Windrush scandal involves British people who are here in this country legally, who have in some cases been denied health treatment, lost their jobs or been threatened with deportation.
While they are not in any way illegal immigrants, it seems that in some cases they have become victims of the "hostile environment" scheme designed to encourage people here illegally to go home.
Labour leader Jeremy Corbyn claimed the Government was warned that its policies would affect the "Windrush generation", a reference to Commonwealth citizens who came to the UK, many of them from Caribbean countries.
Speaking at Prime Minister's Questions, Mr Corbyn said: "Four years ago an internal Home Office memo stated that her hostile environment policies could make it harder for people like the Windrush generation to find homes and, in its own words, 'provoke discrimination'.
"Why did the home secretary ignore that memo?"
But Mrs May pointed out that Labour had also talked about creating a hostile environment.
She said: "'What we're proposing here will, I think, flush illegal migrants out, we are trying to create a much more hostile environment in this country if you are here illegally'.
"Those are not my words. They are the words of the Member for Hodge Hill when he was Labour immigration minister."
She meant Mr Byrne, who is currently Labour's shadow minister for Digital.
The Prime Minister said: "And the Labour leader ought to know about this because the Member for Hodge Hill sits on his frontbench."
Mr Byrne told BirminghamLive: "Mrs May frankly will now try anything and blame anyone to avoid carrying the can for her appalling policy of actually departing British citizens.
"In better days, ministers would resign for a failing like that."
In May 2007, when Tony Blair was Prime Minister, Mr Byrne was the Minister for Immigration and announced plans to force all job seekers to show employers their birth certificate or passport, under plans to root out illegal immigrants.
Bosses who failed to report a worker they suspected was illegal were to be hit with a fines of up to £10,000.
Unveiling the proposals, Mr Byrne said: "We are trying to create a much more hostile environment in this country if you are here illegally."
Companies would have been ordered to check foreign workers every year, forcing them to show ID to prove they were allowed to work in the UK.