The firm reported 221.1 million subscribers across its three streaming platforms at the start of July.
That put it just ahead of Netflix, which has been losing customers.
But Disney warned that its loss of streaming rights for cricket in India would reduce its subscriber growth compared to previous forecasts.
The firm, which also owns general audience television platform Hulu and the sports-focused ESPN+, said demand for its Disney+ product remained strong.
The company added 14.4 million Disney+ subscribers in the quarter, many of them outside of the US - far more than analysts had expected.
Later this year it will launch a new ad-funded service, which will still be charged at the current subscription rate of $7.99. The charge for the ad-free service will rise to $10.99 per month.
The firm plans to launch its ad-funded service outside the US next year.
Executives said they do not expect the rise in prices to put off customers over the long term. The firm is also seeing strong interest from companies hoping to advertise on the new service, they said.
"We are in a position of strength with record upfront advertising commitment," chief executive Bob Chapek told analysts in a conference call to discuss the firm's financial results.
Disney's subscriber gains have come at a hefty cost, with its streaming business losing $1.1bn in the quarter.
Executives said they expect losses to peak this year, In the meantime, a strong rebound in attendance at its theme parks since the worst of the pandemic has provided the firm with a large financial cushion.
Total revenues in the April-June period jumped 26% from last year, pushing profits to $1.5bn.
Shares in the company jumped more than 6% in after-hours trade after the firm shared its results.
Paolo Pescatore, analyst at PP Foresight, called it a "pivotal moment in the streaming wars" saying Disney had more room to grow than arch-rival Netflix.
Netflix lost nearly one million accounts in the most recent quarter, putting its subscriber total at 220.67 million.
"[The results] firmly underline my belief that Disney is at a different phase of growth to Netflix," said Mr Pescatore. "There are still millions of users to acquire as it continues to expand into new markets and rolls out new blockbuster shows."