State labor law exempts federal, state or municipal employees from minimum wage requirements, but Mr. Cuomo’s office said he would act on his own through his Division of Budget, directing the initial wage increases taking effect at the end of year. He would also seek to change the labor law in the coming legislative session, which begins in January, to prevent future governors from using executive powers to roll back the wages. Mr. Cuomo will be up for re-election in 2018, at the end of which the $15 rate will be put into effect in New York City.

The job categories affected by the plan include lifeguards, office assistants and custodial staff, some of which are seasonal. Wages in New York City would be raised first, administration officials said, because of its higher cost of living.

The notion of a statewide $15 minimum wage for all employees, public and private, has faced resistance from Republicans in Albany, and some conservative fiscal groups, who warn it would be disastrous for the state’s economy and job growth. On Tuesday, Senator John J. Flanagan of Long Island, the Republican leader in the State Senate, had no comment on the governor’s decision.

Mr. Cuomo’s announcement was met with an enthusiastic response from labor leaders like George Gresham, the president of 1199 S.E.I.U., the nation’s largest health care workers union, who applauded the governor’s action. Mr. Gresham said it could potentially set a $15-an-hour standard of pay for tens of thousands of the union’s home-care workers, “who do this important work, but who are not necessarily state employees.”

The announcement prompted the city’s public advocate, Letitia A. James, to call for a $15 minimum wage for city employees. A spokesman for Mayor Bill de Blasio, a fellow Democrat and frequent sparring partner of Mr. Cuomo’s, noted that the city had already promised its lowest paid workers more than $12 an hour in 2016, while the governor’s plan would accelerate past that level only at the end of 2017.