Under those new rules, no one under 18 can play; all advertising must show the accurate odds of winning; and operators will have to pay a 15-percent tax on gross revenue made in New York.
Governor Cuomo's office tells CBS News the bill is being reviewed.
Daily fantasy sports sites have come under fire in several states over whether they amount to illegal gambling. The games are based on the real-time performance of athletes on players' draft teams. Unlike traditional fantasy sports that take place over a season, these games are played daily or weekly.
[NOTE: CBS has an ownership stake of less than one-percent in FanDuel.]
On "CBS This Morning," Jason Robins, co-founder and CEO of Draftkings, said he was hopeful that Governor Cuomo would sign the bill into law. "It would be a huge win. New York is a great market, a fantastic sports town, and it's also really been at the center of the debate on fantasy sports over the past six or seven months.
"So having a positive outcome in New York that allows all the fantasy fans around the state, millions of them, to play would be fantastic."
Last November New York Attorney General Eric Schneiderman had issued cease and desist orders to DraftKings and FanDuel. He told CBSN then that "Daily fantasy sports, which we've been looking into for over a month, we've concluded is not some version of fantasy sports. It's just a new version of online gambling."
In the early hours of Saturday, the New York State Senate passed Senate Bill S. 8153 to regulate interactive fantasy sports contests. (A companion bill had previously passed in the State Assembly.)
Robins described following the vote as an "interesting experience" ("I must admit I never thought on a Friday night at 2 a.m. I'd be watching a live webcast from Albany of the New York State Senate!"), and noted that had the bill not passed, it would have been a huge loss for fantasy sports operators.
"I think a lot of eyes around the nation were on New York looking to see what the outcome would be for fantasy sports here," he said.
When asked how much DraftKings has spent in lobbying or the bill, Robins replied, "Honestly, I don't know."
"Tens of millions?" asked Norah O'Donnell.
"No, no, no. Not tens of millions, a lot less than that."
"Two dollars?" asked Gayle King.
"I know it's between two and tens of millions!" Robins laughed. "I do believe the effort that was put in really was the same as a lot of what we're doing in other states.
"At this point I think we have a good understanding of what lawmakers are concerned about. A lot of the consumer protections are very similar in the New York bill as they are in other states. New York is actually the seventh bill this year that a state has passed clarifying that fantasy sports is a legal, skillful activity."
When asked to explain why activities such as DraftKings are not gambling, Robins said it was because it's skill-based: "There's a lot of skill that goes into selecting your teams -- it's like you're the general manager of your team, hence the term fantasy sports. You get to kind of fantasy be the general manager, and you're assembling a team, just like a typical general manager."
"But skill? You're using data to make your choices, but you're still betting on the performance of athletes," said Josh Elliott. "
"And therein lies the debate!" Robins replied.
"Gambling in illegal under the N.Y. State Constitution," said Elliott. "If a consumer comes after you, you could still run head-long into that. ... You could still be forced, for years, to not be able to transact business in the state of New York."
"I don't think that's likely. There are a lot of really smart people -- lawmakers, lawyers -- that weighed in on the language of the bill. ...
"If you look at the majority by which the State Senate and the State Assembly passed [it], I think it shows that a vast majority of the people are very comfortable with the bill, that it's written correctly."
Following the State Senate vote, Attorney General Schneiderman said that, regardless whether the new bill is signed into law, New York State will continues to press its case. "We will nevertheless continue to pursue our claims that DraftKings and FanDuel previously engaged in false advertising and consumer fraud," he said in a statement.
When asked to respond, Robins said, " I think our first step is just going to be to sit down with his office, understand what their concerns are, and make sure that going forward we're addressing them."