From the Wikipedia page 
This is a list of archives in India, which collect and store historical documents, photographs, and other media.
- National Archives of India - established in Kolkata as the "Imperial Record Department" in 1891.
- National Film Archive of India
- National Mission for Manuscripts - established in New Delhi in 2003.
- Archives of Indian Labour
- Bihar State Archives - established in Patna in 1912 as the Civil secretariat record room
- Delhi Archives - established in New Delhi in 1972.
- Delhi Public Library
- Directorate of State Archives (West Bengal)
- Goa Archives - established in Panaji on 25 February 1595 by Diogo do Couto (Wikipedia page ) and was called "Torre do Tombo do Estado da India". The oldest record is from the year 1498.
- Himachal Pradesh State Archives - established in 1979.
- India Office Records
- Karnataka State Archives The new State of Madhya Pradesh came into existence on 1st November 1956. It was carved out of the parts of old Madhya Pradesh (Central Provinces & Berar) the erstwhile States of Gwalior, Indore, Bhopal, Rewa and Madhya Bharat State."
- Kerala State Archives - established in Thiruvanandapuram in 1962
- Madhya Pradesh State Archives
- Maharashtra State Archives - in Mumbai, Pune, Kolhapur and Vidarbha.
- Manipur State Archives - established in Imphal in 1982
- Mizoram State Archives - established in Kohima in 1982.
- Nagaland State Archives - established Kohima in 2012.
- North Bengal State Library
- Odisha State Archives
- Oriental Research Institute Mysore
- Punjab Archives
- Rajasthan State Archives - is established in Bikaner.
- Sikkim state archives
- State Central Library, Hyderabad
- State Central Library, Kerala
- Tamil Nadu State Archives - established in 1909 as the Madras record office.
- West Bengal State archives
A further list can be found here .
- A pdf of The Hand book of The Bombay Archives by Sanjiv P Desai and RS Pednekar, 1978 can be found here