He was seen there with Anant Ambani, son of Reliance Industries chairman Mukesh Ambani, whose Jio 4G service is expected to be a game-changer for mobile internet in India by the end of 2016.
Such high-speed networks would "unleash the power of the iPhone", Mr Cook had said after announcing Apple's results for the first quarter of 2016.
While global iPhone sales fell for the first time ever, a drop of 16% from the first quarter of 2015, they rose 56% in India in the same quarter.
None of the California-based technology giant's seven CEOs ever visited India while in office, though Steve Jobs famously came here as a backpacking hippie looking for "answers" in the mid-1970s.
Apple announced that it would invest $1bn in Chinese ride-hailing app Didi Chuxing last week to "better understand the Chinese market", Mr Cook told Reuters.
The centre would later move to Apple's own campus in or near the city and expand to 2,500 employees, government sources told The Hindu newspaper.