The race's outcome will still be decided by a ranked-choice count but the 33-year-old front-runner remains in a commanding position.
Tuesday night, Mamdani pulled ahead with more than an estimated 80% of ballots counted.
The Democrat ran on an agenda for free buses, healthcare and more affordable housing, which gained traction with voters.
With victory all but assured, Mamdani told supporters, "I will be your Democratic nominee for the mayor of New York City."
"I will be the mayor for every New Yorker, whether you voted for me, for Governor Cuomo, or felt too disillusioned by a long-broken political system to vote at all," he said. "I will work to be a mayor you will be proud to call your own."
Zohran Mamdani declares victory in NYC's Democratic mayoral primary
"We have won because New Yorkers have stood for a city that they can afford. A city where they can do more than just struggle," said Mamdani.
Former governor Cuomo was long considered the favorite throughout the race with his years of political experience, name recognition and fundraising efforts.
Cuomo was looking to make a comeback after the 67-year-old resigned as governor after a sexual harassment scandal.
While announcing he conceded the race, Cuomo told supporters that he called Mamdani to congratulate him.
Cuomo, in a speech to his supporters, said Mamdani "won" and that "we are going to take a look and make some decisions."
"(Mamdani) put together a great campaign and he touched young people and inspired them and moved them and got them to come out and vote. And he really ran a highly impactful campaign," Cuomo said at an event with supporters on Tuesday night.
Cuomo thanked supporters, conceded New York City's Democratic mayoral primary to Zohran Mamdani
Mamdani would be the city's first Muslim and Indian-American mayor, if elected.
Even with Cuomo's announcement, the race is set to proceed to a ranked-choice count since no candidate got more than 50%.
Tuesday's results do not yet reflect any ranked-choice tabulation. More comprehensive results won't be reported until July 1, when the board of elections plans to run the ranked-choice tabulations.
The primary winner will go on to face incumbent Mayor Eric Adams, a Democrat who decided to run as an independent amid a public uproar over his indictment on corruption charges and the subsequent abandonment of the case by Trump's Justice Department. Republican Curtis Sliwa, the founder of the Guardian Angels, will be on the ballot in the fall's general election.
There is also a possibility that Cuomo will run on the November ballot as an independent candidate. Earlier, Cuomo's campaign said he would run on a "Fight and Deliver" ballot line in the general election that he says is meant for voters disillusioned with the Democratic Party.