All it needs for passage is a simple majority. A spokesman for Quinn said she is confident it will get the green light.
"Too many adult smokers begin this deadly habit before age 21," said Quinn in a statement.
"By delaying our city's children and young adults access to lethal tobacco products, we're decreasing the likelihood they ever start smoking, and thus, creating a healthier city," she added.
New York is already strict about tobacco. As in many US cities, smoking is banned in bars and restaurants and parks, but here it is forbidden even on beaches. Cigarette taxes are the highest in the country.
City hall says that in 10 years the percentage of adult smokers has gone from 21.5% in 2002 to 14.8% in 2011. But among young people it has remained unchanged since 2007, at 8.5%.
Authorities hope that raising the legal age for buying cigarettes will reduce the number of smokers aged 18-20 by 55%.
But the move could prove unpopular among those who already feel Mayor Michael Bloomberg should mind his own business.
Late last year he tried to ban super-sized sugary soft drinks on health grounds and many people complained. A judge blocked the measure last month, calling it arbitrary.