The woman said the number of people "flooding" into the UK "who cannot speak English" is costing public services too much during the fiery debate about Home Secretary Priti Patel's new points based immigration proposal.
She has been roundly criticised on Twitter since her comments, which went uninterrupted for about 90 seconds.
The BBC has come under fire for broadcasting and retweeting her views.
Labour councillor for Rochester West Alex Peterson said: "Giving fact-vacuums like this woman a platform to display their ignorance on national TV makes for uncomfortable viewing.
"It’s as manipulative and calculating as when Wogan let George Best loose in the green room then on to his chat show three sheets to the wind."
Anti-Brexit campaign group Best for Britain slammed the BBC for publishing such a debate fearing it could "normalise" the "expression of racist views."
It tweeted: "Truly sad to see such a misinformed audience.
"Broadcasting such opinions to millions of people and framing them as valid, merely normalises the expression of such racist views."
Columnist Tim Walker said the BBC's retweet of the woman's comments was "grotesquely irresponsible" and showed the broadcaster as "actively spreading racist nonsense."
Journalist Peter Geoghegan tweeted: "This is the problem with media as 'just reporting what people say' writ large. Important part of job of journalism is to contextualise statements. This isn't contextualised *at all*, indeed the line about immigrants costing public services too much looks like a statement of fact."
"This is how racism and far right extremism is normalised and legitimised," posted Owen Jones in reply to the programme's tweet. "This is a vile, unhinged rant, packed full of lies and hate, and the BBC decide to uncritically clip it so it can be easily shared across social media."
On the programme Michael Portillo said the UK should not be proud that many of the country's vital services rely on cheap labour coming from abroad.
He said: "There's something rather decadent about the situation where we think there are lots of jobs that only foreigners will do."
A BBC spokesman said: ""Last night’s Question Time included a debate about immigration which featured a broad range of views from the audience members and panellists.
We posted five clips of people expressing their different views on the issue, including a panellist responding directly to the views of an audience member".