The 54-year-old Rajan will also be the group chief marketing officer and chief executive of IDFC Foundation, the bank said in a statement.
Rajan was handpicked by Mistry from consultancy firm EY, and was a member of the powerful group executive council, which was dismantled on the day Mistry was fired, and also served as Tata Sons' nominee on the boards of various group companies.
Other group executive council members, who left the Tatas after the surprising dismissal of Mistry include Nirmalya Kumar, who has gone back to academics and last week joined a Singapore university, and also Madhu Kannan, who has joined as the chief of taxi aggregator Uber India.
"Rajan's diverse experience in industry, consultancy and people management will enable us to navigate the nuances and complexities of a challenging business world," Lall was quoted as saying in the statement.
Rajan is a graduate in economics from Loyola College Chennai, a postgraduate in business management from XLRI, Jamshedpur and holds a doctoral degree from IIT Delhi.