The IT giant said the new centre will focus on developing maps for its products, including iPhones and iPads, and create up to 4,000 jobs, though not all will be direct employees.
It pits Apple against Google, which has made maps a key focus of its attempt to forge inroads in India, a vast but largely untapped Internet market where almost a billion people are still not online.
“The talent here in the local area is incredible and we are looking forward to expanding our relationships and introducing more universities and partners to our platforms as we scale our operations,” Cook said in a statement.
Local government ministers in Telangana state, of which Hyderabad is the capital, expressed their delight on Twitter at the new centre, a partnership with local software firm RMSI.
“Last May it was Google coming to Hyderabad, this May Apple opens its facility in Hyderabad,” K.T. Rama Rao, a Telangana cabinet minister, tweeted.
Maps are a controversial topic in India, which has long-running border disputes with several of its neighbours, most famously over the divided Himalayan region of Kashmir.
A draft bill up for public consultation proposes a billion-rupee ($15 million) fine and jail time for anyone distributing a map the Indian government deems to be “wrong”.
On Wednesday Cook announced the launch of a new app design centre in Bangalore after jetting in from China.
He is expected to meet with India’s tech-loving Prime Minister Narendra Modi in the coming days.
In April Apple reported its first drop in global iPhone sales since launching the smartphone in 2007.
It faces a stiff challenge in India, where high taxes take the cost of a basic iPhone to almost $600 and the vast majority of consumers gravitate to cheaper brands.
During his trip to China, Cook announced Apple had invested $1 billion in Chinese ride-hailing app Didi Chuxing, rival of US-based Uber.