The motion, tabled by councillor Paul Blanchard, described the production of foie gras as "a cruel practice".
The delicacy is now banned from council premises and the council will write to hotels and restaurants in the city to inform them of its opinion on the subject.
The move makes York the first city in the UK to call for a nationwide ban on the sale of foie gras on cruelty grounds.
Prior to the vote, animal rights campaigners were demonstrating outside the city's full council meeting.
Blanchard, a Labour councillor, originally wanted York to emulate Chicago, by passing a total ban on the sale of foie gras in the city, but legal advice to the council said that was not possible.
The alternative motion stated: "This council believes that this intolerably cruel and painful practice is unnecessary and should end.
"The city of Chicago, USA, has shown tremendous compassionate leadership in their brave decision to ban foie gras in its shops and restaurants.
"The torture of small innocent animals should not be a matter of personal dietary choice."
The council's chief executive will now write to Lord Rooker, the minister for sustainable food and farming and animal health, stating the authority's concern over the sale of foie gras in the UK.
He will also request a review of central government policies on animal welfare issues.