BJP activists took out a mass rally from Ram Mandir Square here and marched towards the chief minister's residence with banners, party flags and placards.
They raised slogans against Mr Patnaik and demanded his resignation over the issue.
A scuffle ensued with the police when they reached near the chief minister's residence 'Naveen Niwas' and attempted to enter it forcefully, police said.
A large number of BJP workers were then detained, the police added.
The girl had alleged rape by four persons in uniform when she was returning home from Kunduli market in Koraput on October 10 last year and committed suicide on January 22.
Odisha police had rejected the girl's allegation and said she was not raped.
The state government has constituted a judicial panel to probe into the alleged gang rape and subsequent suicide by the victim.