Mr Kumar fought the polls in a "grand alliance" with former rival Laloo Prasad Yadav  and the Congress party.
Mr Yadav's son Tejaswi Yadav was sworn in as the deputy chief minister.
The ceremony was attended by regional parties from across India, some of which will face the BJP in state polls.
West Bengal Chief Minister Mamata Banerjee, Delhi Chief Minister Arvind Kejriwal, and Karnataka Chief Minister S Siddaramaiah were among those in attendance.
The win by the "grand alliance" has given the opposition a major shot in the arm - the chief minister of India's most populous Uttar Pradesh state has already said he would consider forging similar alliances to contest the regional election there in 2017.
Prime Minister Narendra Modi led the BJP's campaign in Bihar and often attacked Mr Kumar and his coalition partner, Mr Yadav.
The polls were seen as a test of Mr Modi's popularity, and vital to ensure continuance of his economic reforms policy.
But the grand alliance of regional parties took 178 seats of the 243 that were contested. The BJP only managed to win 58.
Mr Modi and BJP president Amit Shah had raised the sensitive issue of cow slaughter and consumption of beef during the campaign - analysts say these were among the factors that alienated a large number of voters.
Nitish Kumar has been sworn in as the chief minister of the northern state of Bihar after defeating Prime Minister Narendra Modi's BJP in a bitterly contested election.