West Bengal Chief Minister Mamata Banerjee, cabinet members of both the countries, academics and cultural representatives were also present at the event on Friday.
The unveiling ceremony was followed by the convocation ceremony of the university.
After the formalities of inauguration of the building, the two prime ministers were expected to hold a bilateral meeting.
Funded by the Bangladesh government, the building has a 450-seat theatre, the largest of the university.
A library of books on Bangladesh and its relations with India is also there besides a collection of historical and cultural artifacts of Bangladesh.
Murals of the Father of the Nation Bangabandhu Sheikh Mujibur Rahman and Rabindranath Tagore have been placed on the two sides of the building’s entrance.
Hasina announced the construction of the Bangladesh Bhavana during the joint celebrations of Rabindranath’s 150th birth anniversary in 2011.
From the Visva-Bharati, she will travel to Kolkata on a helicopter and stay at the Taj Bengal hotel for the night.
She will visit Rabindranath’s birthplace, the Thakur Bari, at Jorasanko and meet business leaders of Kolkata at the hotel in the evening.
On Saturday, she will travel to Asansol to join the special convocation of the Kazi Nazrul University.
Hasina will also receive honorary D Lit degree from the university.
West Bengal Chief Minister Mamata Banerjee will also address the convocation.
On her return to Kolkata from Asansol, she will visit the Netaju Museum in the afternoon and return to Dhaka later in the night.
The prime minister visited India for the last time in April last year.