The city estimated that more than 4,000 workers to whom the law will apply over the next five years are employees in retail or fast-food businesses, earning close to the state’s minimum wage of $8 an hour. For such workers, the change will increase annual income by about $10,000.

A more sweeping increase in the city’s minimum wage, one of Mr. de Blasio’s top goals in the coming months, will have to wait — but perhaps not for long. Mr. de Blasio, whose administration is working to help Democrats capture a majority in the State Senate, said he hoped the Legislature would take up the issue early next year.

Gov. Andrew M. Cuomo, who in February said that allowing local governments to set their own minimum wages would yield “a chaotic situation,” seemed to have reversed himself months later. He said he would support a plan, advocated by the Working Families Party, that allowed municipalities with higher costs of living to set their own minimum wages.

As a result, the governor has endorsed an increase to $10.10 in the statewide minimum wage, with a provision allowing New York City and other areas to raise their minimums as much as 30 percent higher, to $13.13.

The shift by Mr. Cuomo was among several highlights this year for advocates of a higher minimum wage, which began with President Obama’s support for a national minimum wage of $10.10 an hour. In June, Seattle became the first large city in America to approve a minimum wage of $15 an hour. Other cities, including Chicago and Los Angeles, have also advanced plans to increase their minimums.

It was this national momentum, Mr. de Blasio said, that altered the administration’s plan to expand the living wage law by the end of February. The administration said it had at first expected only to broaden the scope of the law to cover more workers, not raise the wage itself.

“We originally anticipated building more narrowly upon the previous legislation,” Mr. de Blasio said. “When we looked closely, we decided we needed to do a bigger overhaul.”