Up to 10 million more pilgrims are set to do so by the end of the day.
The event, every 12 years, is billed as the biggest gathering on Earth. More than 100 million people are expected to attend the 55-day festival.
Hindus believe a festival dip will cleanse sins and help bring salvation.
In 2001, more than 40 million people gathered on the main bathing day of the festival, breaking a record for the biggest human gathering.
The Kumbh Mela has its origins in Hindu mythology – many believe that when gods and demons fought over a pitcher of nectar, a few drops fell in the cities of Allahabad, Nasik, Ujjain and Haridwar – the four places where the Kumbh festival has been held for centuries.
The festival formally started at dawn on Monday.
Led by naked ash-smeared men with marigold garlands around their necks, Naga sadhus (ascetics) arrived in a colorful procession and sprinted into the chilly waters of Sangam – the point at which the rivers converge.
Teams are managing crowds on the river bank – as soon as pilgrims finishing bathing, they are encouraged to move away and make space for other bathers.
“I have washed off my sins,” Mandita Panna, a resident of Nepal and an early bather, said.
Allahabad has been preparing for the festival for months and a vast tented city has grown up around the river.
Fourteen temporary hospitals have been set up with 243 doctors deployed round-the-clock, and more than 40,000 toilets have been built for the pilgrims.
Police checkpoints have been set up on all roads leading to Allahabad and about 30,000 policemen and security officials have been deployed to provide security during the festival.
Tens of thousands of men, women and children have set up camp on the white sands of the river front.
On Sunday night, smoke could be seen rising from hundreds of small fires which people had built to cook dinner or keep warm.
The main attraction at the festival is the Sadhus – Hindu holy men – who have been leading processions accompanied by elephants, camels, horses, chariots and music bands in recent days.
The festival has prompted health concerns, however, with campaigners warning that the river waters are heavily polluted.
Most pilgrims drink a few drops of the Ganges water and many fill bottles to take home with them.
Authorities say they have taken steps to address the concerns.
Last week, companies along the banks of the Ganges and Yamuna were warned against discharging any pollutants into the waters.
Reservoirs upstream have been ordered to discharge fresh water into the rivers ahead of the six big bathing days, and the festival authorities have declared the Kumbh Mela area a plastic-free zone.
The Kumbh Mela, which is costing the authorities 11.5 billion rupees ($210 million) to organize, is expected to generate businesses of at least 120 billion rupees, according to a report by industry association The Associated Chambers of Commerce and Industry (Assocham).
The report says that the festival is expected to draw over a million foreign tourists too.
Kumbh Mela Festival in numbers:
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