"The demand for Disney+ has exceeded our highest expectations," the company said. "We are so pleased you're excited to watch all your favorites and are working quickly to resolve any current issues. We appreciate your patience."
Thousands of complaints were flagged Tuesday morning, according to Downdetector, a website that collects complaints about websites and apps.
Disney spokeswoman Jessica Casano declined to say how many users experienced problems and how many successfully streamed content.
Disney chief executive Bob Iger set huge expectations for Disney Plus in his 2017 announcement for the service. "We're going to launch big, and we're going to launch hot," he said.
Splashy debuts have bedeviled streaming services for years. HBO streaming buckled under the enormous demand on servers after April's final season premiere of "Game of Thrones," and the ESPN Plus app has had fits while streaming some UFC fights.
Disney learned from those problems. In the lead-up to the launch, Disney executives said they prepared for this exact problem of huge but predictable demand.
"We've built capabilities to sustain that, both in terms of processing the transactions that all come in - in a very, very short period of time - as well as the streams," Michael Paull, the company's head of streaming, told the Verge in August.
Disney Plus costs US$6.99 ($11.04) a month, and a bundle of Disney Plus, Hulu and ESPN Plus will set you back US$12.99 a month.
The service is available in the United States, Canada and the Netherlands, the company said, and will launch in Australia and New Zealand on November 19, though it is unclear whether technical problems will persist through the week.
That second round of releases will also include Puerto Rico. Casano declined to explain why it wasn't available there for the US release Tuesday. Puerto Rico is a US territory.
The rocky premiere of Disney Plus comes as the latest front in an escalating streaming war for consumers' eyeballs. Apple TV Plus launched November 1, taking on Amazon, HBO, Hulu and Netflix. (Amazon chief executive Jeff Bezos owns The Washington Post.)