The low-intensity blasts occurred in carriages of an overnight train from Bangalore as it pulled into Chennai central station shortly after 7:00 am.
A 22-year-old woman working at IT outsourcing firm TCS was killed when one of the devices which had been placed underneath her seat exploded, said Rakesh Misra, general manager of the southern region for Indian Railways.
"Civil police are investigating what kind of bomb it was and why anyone would have placed the bomb," he told reporters at the scene.
Another 14 people were injured, Chennai police chief J.K. Tripathi told AFP, at least two of them seriously.
Security forces cordoned off the blood-splattered platform where a large crowd of onlookers and delayed passengers gathered, while searches by bomb detection teams were stepped up in other stations.
Tripathi told AFP that they were yet to detain any suspects, despite local media reports of one or two people being questioned over the crude explosives.
Prime Minister Manmohan Singh condemned the attack, saying that "such barbaric acts targeting innocent men, women and children only highlight the desperation and cowardice of those responsible".
Multiple low-intensity explosions are typically the hallmark of the Indian Mujahideen, a home-grown militant network which is thought to have been weakened by the recent arrests of senior figures.
Sixteen people were killed in February 2013 when bombs strapped to bicycles exploded in the southern city of Hyderabad in an attack blamed on the network.
India's election, which ends on May 16 with results, has seen repeated attacks by Maoist rebels based in the forests of central India who have killed at least 25 so far, according to an AFP tally.
One panic-stricken passenger told the Press Trust of India (PTI) news agency that he saw three to four people "writhing in pain" after the blast on the train, which was set to terminate in the remote northeastern city of Guwahati.
The injured have been taken to the Rajiv Gandhi Government General Hospital in Chennai.
The dead woman was identified as an employee of Bangalore-based TCS, with the firm saying it was "deeply shocked and saddened at this unfortunate loss of life of our colleague in this tragic incident".
"Our thoughts and prayers are with the family at this time," said a spokesperson.
Her grandmother, named as Rajalakshmi, told PTI that she was travelling home and was set to marry in another two months.
"She was to come home today. But now only her body is coming", she was quoted as saying.
Security has been tightened across India as the country holds its six-week general election.
Voting has already taken place in most major cities, including Chennai which went to the polls last week.
K. Ramanujam, police director general in Tamil Nadu state, of which Chennai is the capital, said the city may not have been the target of the blasts because the train was running late.
"It is premature to say what kind of device was used in the blast. Damage to the train is not heavy," he said.
Tamil Nadu Chief Minister Jayalalithaa Jayaram "strongly condemned" the attacks and urged people not to panic.
Families of the victims will receive compensation, Railways Minister Mallikarjun Kharge told reporters.
The train continued its journey towards the city of Guwahati in the northeastern state of Assam once the three damaged coaches were replaced.
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@Thank God Im Not An Indian: Thank you very much !!! for god you are not indian
@Thank God Im Not An Indian: Pot calling the kettle black? If chances given, even Muslim Pakistanis would leave the land of pure and live in India, because they know, there are daily blasts in Pakistan with the population of 18crore, but India had one blast in months killing unfortunately 2, despite being a nation of 1.25billion. Find per killed ratio yourself. :p
India Shining!
And we so know who is behind it. iM is an offshoot of we know who