The proposal, which was championed by Gov. Andrew M. Cuomo, was supported by 57 percent of voters, with most precincts reporting.

The approval is a milestone in the gambling industry’s long, expensive push to tap into the New York market, an effort that has spanned decades, cost tens of millions of dollars and is certain to continue as gambling companies vie for the right to develop the new casinos.

New York State already has five Indian-run casinos, all of them upstate, and nine slot machine parlors at racetracks. And the State Legislature, at the urging of Mr. Cuomo, has required that at first, only four new casinos will be permitted, and only upstate: in the Albany area, the Catskills-Hudson Valley region and part of the Southern Tier, a region along the northern border of Pennsylvania.

Mr. Cuomo, a Democrat, has said that casino development would create jobs, lure tourists and allow the state to recapture some of the gambling revenue now flowing to nearby states with casinos.

“We literally hemorrhage people from the borders who go to casinos,” Mr. Cuomo told reporters after voting in Westchester County on Tuesday. “I think it will keep the money in this state, and I think it’s a major economic development vehicle for the Hudson Valley especially and for upstate New York.”

Voters who marked their ballots in support of the proposed amendment said the potential financial gain was too significant for the state to pass up, especially given the slumbering economy in parts of upstate.

“We really need something to come in and revitalize this area,” Jackie Burris, a retired special education teacher who has a second home in the Catskills hamlet of Rock Hill, said after voting in Great Neck. “We need it desperately.”

New York’s debate over casinos was far more muted than in other places where casinos have been proposed, in part because Mr. Cuomo drafted the casino plan in a way that neutralized many likely opponents, like the Indian tribes that run the upstate casinos.

The casino amendment was promoted by a coalition called New York Jobs Now, which was primarily financed by gambling interests, including operators of the racetrack slot machine parlors, as well as labor unions that stood to benefit from the development of casinos. The coalition raised more than $4 million for the effort, which included a series of television ads promoting the economic benefits of casino development and citing endorsements from newspaper editorial boards.

The supporters of the measure benefited from the wording on the ballot, which portrayed the proposal in a positive light, suggesting that it would lead to new jobs, increased school financing and property tax relief, and omitting any negative arguments. The wording, adopted by the State Board of Elections after consultation with Mr. Cuomo, drew criticism from government watchdog groups, but a lawsuit challenging it was unsuccessful.

A disparate collection of opponents tried to persuade voters to oppose the casino expansion, but their efforts were largely homespun. One group of opponents paid for a last-minute, less-than-polished television commercial quoting former Gov. Mario M. Cuomo denouncing casino gambling two decades ago, but opponents generally had little money to back up their efforts. (In response to the ad, Mario Cuomo endorsed the referendum, saying, “A great deal has changed in 20 years.”)

The opponents argued that Andrew Cuomo and other supporters of the proposal were being overly optimistic about the economic benefits, citing the struggles of destinations like Atlantic City as casinos have proliferated on the East Coast.

The state’s Catholic bishops urged voters to “consider the potential for negative consequences” and the Episcopal bishop of New York warned against “false hopes,” saying, “In places where casino gambling has been introduced, almost all gains have come at the high social cost of addiction and family disintegration, and deepening poverty.”

The push to expand gambling was a textbook case of how well-financed interests can push Albany to embrace an industry, hiring an army of lobbyists and wooing Mr. Cuomo and lawmakers with millions of dollars in campaign contributions.

The new casino licenses will be awarded by a panel to be appointed by the State Gaming Commission. Developers have expressed the most interest in the Catskills, where large resorts once prospered.