The cartoon shows the couple standing in the presidential Oval Office in front of a fireplace in which an American flag is burning.
The illustration by Barry Blitt showed Obama in sandals, robe and turban, and his wife dressed in camouflage pants, combat boots and an assault rifle.
The New Yorker issue, titled The Politics of Fear, hit the newsstands on Monday.
The presidential candidate did not respond when asked about the cartoon, but Bill Burton, a campaign spokesman, condemned the cover, saying the magazine might think of it as “a satirical lampoon of the caricature Senator Obama’s right-wing critics have tried to create, but most readers will see it as tasteless and offensive, and we agree”.
John McCain, the Republican presidential candidate, said the cover was “totally inappropriate”.
The New Yorker said the drawing “combines a number of fantastical images about the Obamas and shows them for the obvious distortions they are”.
“The burning flag, the nationalist-radical and Islamic outfits, the fist-bump, the portrait on the wall? All of them echo one attack or another,” it said on Monday.
“Satire is part of what we do, and it is meant to bring things out into the open, to hold up a mirror to prejudice, the hateful, and the absurd. And that’s the spirit of this cover.”
The magazine added that there were also two “very serious” articles on Obama in the same issue.
Patrick Egan, an assistant professor of politics at New York University, said people who read The New Yorker would see the cover’s satirical character as obvious.
“Anyone who knows anything about the New Yorker is going to be able to recognise that this is satire,” he told Al Jazeera.
“It’s taking what some Americans falsely believe and blowing it up to literally cartoonish proportions. And, in that sense, it is sophisticated satire and something that a lot of people can kind of laugh over.”
The cartoon controversy has highlighted the misconceptions about Obama.
A recent Newsweek magazine poll showed that more than a quarter of the American electorate believe that Obama was raised a Muslim and 12 per cent believe he remains a practising Muslim.
Obama is a Christian, and his campaign has had to set up what it calls an anti-smear website to combat rumours spreading on the internet about his faith and character.