Police say the blasts ripped through a roadside food stall and an amusement park in Hyderabad late Saturday within minutes of each other.
Hundreds of people were at the two popular spots.
A laser show was going on in the amusement park when the explosion ripped through it.
People who escaped the attack say seats were hurled high into the air by the force of the blast. The food stall was destroyed.
The chief minister of the state, Y.S. Rajasekhara Reddy, called the blasts a terrorist attack, and he urged people to remain calm.
The Andhra Pradesh Home Secretary, K. Jana Reddy said it was one of the worst attacks in the city. "It is a very sad and bad incident in the history of Hyderabad. My government condemns it vehemently," he said.
Indian Home Secretary Madhukar Gupta said it is too early to say who was responsible. "Whatever has happened needs to be condemned and looked at and dealt with all the seriousness that is called for. All further enquiries and so on will be made," he said.
Police say they later found two unexploded bombs at other spots in the city.
Hyderabad is one of India's biggest cities and a key information technology hub. It has a large Muslim population, and a history of communal clashes between Hindus and Muslims.
However, Saturday's blasts went off at spots frequented by both Hindus and Muslims. This is the second time the city has been hit by such attacks. In May this year, 11 people were killed in a bombing at a mosque.
Other Indian cities have witnessed similar attacks. The worst took place in Mumbai last year when a series of bombs ripped through the city's commuter train network, killing about 200 people.
Police and security experts have blamed past attacks on Pakistan-based Islamic militant groups.