Bombay High Court Chief Justice Mohit Shah administered the oath of office to Sinha.
Goa Chief Minister Manohar Parrikar and other dignitaries were present at Raj Bhawan here on the occasion.
Goa had no full time Governor following Wanchoo's resignation from the post after his role was questioned in VVIP chopper deal.
Sinha, 71, becomes the first woman Governor of the state with a full-time charge.
Born on November 27, 1942, at Chhapra village in Muzaffarpur district of Bihar, Sinha is a renowned Hindi writer and former chairperson of Central Social Welfare Board (CSWB), Ministry of Human Resource Development.
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She had actively participated in 'Samagra Kranti' led by Jai Prakash Narayan.
After taking a post graduation degree in psychology, she completed her B.Ed. She started as a college lecturer at Dr S K Sinha Women's College, Motihari in Bihar and later became principal of Bharatiya Shishu Mandir at Muzaffarpur.
Sinha is married to Dr Ram Kripal Sinha, who was earlier a college lecturer and later became a Cabinet minister in the Bihar state government.
She has written more than 46 books in various disciplines. Her literary contributions include 'Jyon mehandi ko rang' (novel - a short film on Doordarshan was made based on the same), 'Nayi Devyani' (novel), 'Gharwaas'(novel), 'Sita puni boli' (novel) and many more.
Her story collections include 'Sakshatkar', 'Ek Diye Ki Diwali', 'Apna Jeevan' etc. She has also written essays like 'Aine ke saamne', 'Maanvi ke Naate' and 'Bihar Ki Lok Kathaye' for children.
She contributes regular columns in various national and regional Hindi newspapers, magazines and periodicals.