The accident occurred late Sunday after one of the passenger trains overshot a signal between the towns of Alamanda and Kantakapalle in Andhra Pradesh state, AFP reported.
A preliminary investigation found that "human error" had led to the collision, the railway ministry said in a statement.
"Thirteen passengers have been killed and 50 others are injured. Rescue operations are going on," top local government official Nagalakshmi S. told reporters.
Indian Prime Minister Narendra Modi said he had spoken with the country's railway minister about the "unfortunate" derailment.
"Authorities are providing all possible assistance to those affected," Modi wrote on social media.
India has one of the world's largest rail networks and has seen several disasters over the years, the worst in 1981 when a train derailed while crossing a bridge in Bihar state, killing an estimated 800 people.
In June, a three-train collision killed nearly 300 people in Odisha state.
In August, at least nine people were killed when a coach parked in southern India caught fire as a passenger tried to make tea.
And earlier this month, four people were killed after an express train derailed in Bihar.