Bhaskar was the daughter of naval officer Chitrapu Uday Bhaskar. She was born and raised in Delhi. After receiving her bachelor's degree in English literature from the University of Delhi, she enrolled herself at the Jawaharlal Nehru University to complete her master's in Sociology. Bhaskar made her film debut with a supporting role in the 2009 drama Madholal Keep Walking, a commercial failure. She achieved wider recognition for her supporting role of a bride-to-be in the commercially successful romantic comedy Tanu Weds Manu (2011). Her performance in the film was well received by critics and she garnered a Filmfare Award for Best Supporting Actress nomination.