The Prime Minister reacted furiously to Mr Corbyn's strong criticism of the Government's treatment of the "Windrush generation", who have faced challenges to their immigration status.
Mrs May pointed to a Commons debate which saw standing ovations and applause for Labour MPs who shared stories of the anti-Semitic abuse and threats they have received, including from those purporting to support Mr Corbyn, and their fears of increasing hostility towards Jewish people.
Speaking during Prime Minister's Questions, Mr Corbyn launched a final attack on the Government's immigration policy by telling MPs: "This is a shameful episode and the responsibility for it lies firmly at the Prime Minister's door.
"Her pandering to bogus immigration targets led to a hostile environment for people contributing to our country.
"It led to British citizens being denied NHS treatment, losing their jobs, homes and pensions, thrown into detention centres like criminals and even deported, vital historical records shredded and ministers blaming officials.
"The Windrush generation came to our country after the war to rebuild our nation that had been so devastated by war.
"Isn't the truth, Mr Speaker, that under her the Home Office became heartless and hopeless?
"And doesn't she now run a Government that is both callous and incompetent?"
Mrs May said the "Windrush generation" helped to build the country, worked in public service and have a right to be here as they are British, with efforts ongoing to ensure people have the documentation they need.
She added: "(Mr Corbyn) talks about being callous and having a disregard for people.
"I have to say to him that I am the Prime Minister who initiated the Race Disparity Audit, which said 'What are we doing in this country to ensure people have equal opportunities in this country?'.
"And can I say to (Mr Corbyn), he talks about being callous - I will not take that following a debate last night where powerful contributions were made, particularly by the members for Stoke-on-Trent North (Ruth Smeeth), Barking (Dame Margaret Hodge) and Liverpool Wavertree (Luciana Berger).
"I will not take an accusation of callous from a man who allows anti-Semitism to run rife in his party."
The Prime Minister later paid tribute to Ms Smeeth and Ms Berger for their "incredible bravery" in speaking out.
It came as Tory former leader Iain Duncan Smith described their speeches in Tuesday's debate as "deeply moving", adding: "They were both horrifying in the sense of the abuse that they have faced, but also uplifting in the sense of the bravery that they have shown in tackling their abusers.
"Can I ask (Mrs May), does she not agree with me, what came out of that debate yesterday was that not only should every political party have absolutely no place for anybody who is an anti-Semite, but also, just as importantly, should kick out of the party any apologist for anti-Semites?"
Mrs May said: "I think it is incredibly important for us in political parties in this country to show a very clear signal that we will not accept, we will not tolerate anti-Semitism in any form.
"I made reference to a number of speeches that were made yesterday. Can I also join him in commending those members of this House, particularly (Ms Smeeth) and (Ms Berger), who have suffered incredible abuse as a result of this anti-Semitism, but who also have shown incredible bravery in being willing to stand up and set that out to this House.
"Theirs was a fine example of the best of this House of Commons and the best of Members of Parliament."