"I started my life journey from a small tribal village," Murmu, 64, said after taking the oath of office in parliament.
"From the background I come from, it was like a dream for me to even get elementary education," she added.
"But despite many obstacles, my resolve remained strong and I became the first daughter from my village to go to college."
Murmu's win was considered a certainty because of the strength of the ruling Bharatiya Janata Party and its allies in the parliament and state assemblies.
Analysts said the move will likely help prime minister Narendra Modi extend his base among the poor tribal communities ahead of his re-election bid in 2024.
"Her assuming the Presidency is a watershed moment for India especially for the poor, marginalised and downtrodden," Modi said on Twitter after Murmu's address.
Murmu said her election would give hope to those left behind by India's recent economic growth.
"It is a matter of great satisfaction to me that those who have been deprived for centuries, who have been away from the benefits of development... are seeing their reflection in me," she said.
India's prime minister wields executive power, but the president can send back some parliamentary bills for reconsideration and also plays a guiding role in the process of forming governments.
Murmu is the country's second woman president after Pratibha Patil, who held the position for five years from 2007.
She succeeds Ram Nath Kovind, the second president from the Dalit community, the bottom of the Hindu caste system.