Patil was sworn in at a simple function in Raj Bhawan.
Rajasthan High Court Chief Justice Jagdish Bhalla administered the oath of office.
Rajasthan Chief Minister Ashok Gehlot, cabinet ministers and senior state bureaucrats attended the swearing-in ceremony.
Two minutes of silence was observed for Rau, who passed away on Monday, Apr 26 after suffering a heart attack.
Rau was the fourth Rajasthan governor to die in office. In fact, she took over as governor in Jan 2010, after her predecessor SK Singh passed away while in office in Dec 2009.