Koshyari, 78, who was administered the oath by Chief Justice of Bombay High Court Justice Dipankar Datta, will continue as Maharashtra Governor with additional charge of Goa.
He succeeds Satya Pal Malik who was on Tuesday transferred from Goa and appointed Governor of Meghalaya.
Koshyari read out his oath in Konkani. Goa Chief Minister Pramod Sawant and Leader of Opposition in Assembly Digambar Kamat were among those present on the occasion.
Koshayri also received a guard of honour from Goa police.
An official release said Koshyari,who assumed the office of the Governor of Maharashtra in September 2019 brings with him a vast and varied experience of public service.
In 1997, Koshyari became a member of the Uttar Pradesh Legislative Council and after Uttarakhand state was formed, he became minister of energy, irrigation, law and legislative affairs, the release said.
Koshayri was the chief minister of Uttarakhand during 2001-02. He was elected to the Rajya Sabha from Uttarakhand in November 2008 and held the position till 2014, it added.
Meanwhile, Malik left Goa in the morning to take charge as Governor of Meghalaya.