The Facebook-famous Winklevoss twins donated $100,000 to Gov. Cuomo’s campaign — and less than a month later, state authorities approved their cryptocurrency exchange, campaign-finance filings show.

And once the Department of Financial Services gave the stamp of approval, the twins chipped in another $30,000 for the governor’s re-election.

The big-bucks donations were the first checks that Cameron and Tyler Winklevoss ever cut for Cuomo and account for $130,000 of the $140,900 they gave to candidates in the state.

The men sent identical $50,000 donations on April 24.

On May 14, state regulators licensed the Winklevosses’ firm, Gemini Trust Company, to trade Bitcoin and other emerging currencies.

Gemini was the “first qualified custodian and exchange” to offer trading of the “emerging digital currency Zcash in New York,” officials bragged in the press release.

Then, a month later, the twins were dipping into their bank accounts again to each give Cuomo another $15,000 on June 20.

Everyone involved insisted there was no quid pro quo.

“We appreciate their support,” said Cuomo reelection-campaign spokeswoman Abbey Fashouer. “No contribution of any size influences any government action.”

A Winklevoss rep defended the donations. “Tyler and Cameron Winklevoss have supported — and continue to support — many effective leaders around the country,” the spokeswoman said.

“They contributed to Gov. Cuomo’s campaign because they believe he’s doing a great job, is a problem-solver, and is definitely the right person to lead New York in the years ahead.”

But this was the first time in Cuomo’s two terms as governor that the twins have supported him financially.

State financial officials said they did nothing wrong in approving the license.

“DFS takes care in reviewing every application,” said spokesman Richard Loconte.

“The department has no knowledge of political contributions and such contributions have absolutely no bearing on DFS’s independent authority and decision making.”

Officials also approved a license for a Winklevoss competitor, itBit, in May.

The trading license could boost the legitimacy of the Winklevosses’ ZCash currency, which is worth just a fraction of Bitcoin.

Its market cap is currently $878 million, while Bitcoin is valued at more than $122 billion.

The mega-donations weren’t the end of the twins’ support for the Democratic incumbent, who faces a spirited primary challenge from actress and activist Cynthia Nixon.

Sources told The Post’s Page Six the twins were named to the host committee for Cuomo’s July 14 fundraiser at the Montauk, LI, hot spot Surf Lodge, where Tiffany Trump and Malia Obama have partied this summer.

The minimum donation to be named as a host was $1,000 per person, sources said.

According to campaign records, the twins have contributed to just one other state politician, former state Sen. Dan Squadron (D-Manhattan), who received $10,900 in 2012 and 2014.

Cuomo reported raising another $6 million during the first half of the year and has more than $31 million in cash on hand.

His challenger, Nixon, reported raising $1.6 million since entering the race in March and has about $660,000 in the bank.

Additional reporting by Kevin Dugan