The incident occurred around 2000 local time (1430 GMT) on two platforms as passengers waited to board trains to Prayagraj city, where the festival is being held, news broadcaster NDTV said, Reuters reported.
The death toll included 10 women and three children, local media said.
"People were running across platforms and there was a chaotic situation that led to people falling on each other," a man who saw the stampede told ANI.
Delhi's chief minister Atishi, who only uses one name, told reporters 15 people had died while NDTV reported that the death toll had hit 18 people. She said on X that many of the victims were pilgrims on their way to the festival.
"Distressed by the stampede at New Delhi Railway Station. My thoughts are with all those who have lost their loved ones," Indian Prime Minister Narendra Modi said on X.
Authorities ordered an inquiry into the incident and said the situation was now under control.
Dozens of people were killed in a pre-dawn stampede at the festival - which is held once every 12 years - last month as tens of millions of Hindus gathered to take a dip in sacred river waters on the most auspicious day of the six-week festival.
The Hindu festival has been expected to draw some 400 million people overall, according to officials.
India has witnessed several rail accidents in the last two years, including a collision in 2023 that killed at least 288 people. Indian railways is the fourth largest train network in the world and is undergoing a $30 billion upgrade, as part Modi's push to boost connectivity.