"It is going really well," Cook during an on-stage chat on the opening evening of a Wall Street Journal technology forum on the Southern California coast.
"Lots of people are liking it."
People have begun ending free trials of the music service, which launched at the end of June in more than 100 countries.
More than eight million people are still in the free trial of Apple Music, pushing the total number of users above 15 million, according to Cook.
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He credited a human curation element - actual people who fashion playlists - for creating listening experiences superior to that delivered by "zeroes and ones" of computer algorithms.
"We have music experts just like the DJs when we were growing up," Cook said, setting the service apart from entrenched rivals such as Spotify and Pandora which use software to tailor tunes to people's tastes.
Apple Music, the tech giant's new streaming service, went live at the end of June as the company behind iTunes looks to dominate the fast-growing sector.
Apple Service began with the launch of Beats 1, an international radio station that will feature shows by high-profile artists, and offered streaming - for the first time - of Taylor Swift's blockbuster "1989" album.
To edge its way into the streaming music market, Apple has offered a three-month trial period to new subscribers, after which subscriptions cost $9.99 per month.
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On another entertainment front, Apple will begin taking orders for new Apple TV hardware beginning on October 26 and shipments will start by the end of that week, according to Cook.
"I think it will be disruptive of the TV watching experience," Cook said.
"This is the foundation of the future of TV."
The new Apple TV unveiled last month has the potential to do for television what iPhone did to mobile phones, while claiming a starring role in home entertainment.
Updated Apple TV hardware was not expected to revolutionize the television industry, but it could strike a blow to cable companies that have been in a power seat when it comes to delivering shows and other content.
The new Apple TV will have a version of the App Store that has been a hit on iPhones.
Siri virtual assistant software built in Apple TV allowed for natural language searches for shows, such as asking for something funny or a certain actor by name.
People should also be able to see what they want on-demand instead of being at the mercy of cable broadcast schedules.
These options can spur a trend of "cord cutting" or ending the subscription "bundles" offered by cable and satellite TV firms.
By letting media companies keep control of their content in apps, Apple could find new money-making models while sidestepping worries studios might have about distribution rights.
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"What has to happen in the TV land is it has to be brought up and modernized," Cook said.
"It is almost as though you step into a time capsule when you step into your living room."
The new Apple TV will launch with a starting price of $149.
Apple TV has lagged behind rivals such as Roku and Google Chromecast.
Cook declined to provide figures regarding Apple Watch sales, but said the California company shipped "a lot" in the first quarter they were released and that number has ramped each subsequent quarter.
He dodged questions regarding what kind of plans, if any, Apple had for making a car.
Rumors have abounded in recent months of Apple working on a self-driving car, in its own spin on work being done by Google and Tesla.
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