At least 15m people are expected on the first day alone. Officials project about 120m visitors over 49 days.
Hindus believe bathing at the rivers will cleanse their sins and bring salvation.
The holy men were among the first procession to arrive early Tuesday.
The BBC's Geeta Pandey, who is at the scene, said the men, known as sadhus - or ascetics - smeared ash on their bodies as they came out of the water and chanted "Har Har Gange", or "Mother Ganges", and danced while posing for photographers.
More than a million foreign pilgrims will also take part in the festival, senior administration official Rajeev Rai told the BBC.
He and other organisers have been preparing for more than a year for the event, which dwarfs the annual Hajj pilgrimage to Islam's holiest sites in Saudi Arabia.