Two persons, including the wife of a BJP activist, have been detained in this regard, they said.
The attack took place during the inauguration of Talsari beach festival under Bhograi block in Balasore district. None was injured, police said.
Local MLA and Higher Education Minister Anant Das, Tourism and Culture Minister Ashok Chandra Panda and Korei MLA Akash Das Nayak were also present at the event.
"This is an unfortunate incident. We have detained two persons including a woman," Niti Sekhar, Balasore Superintendent of Police told reporters.
The woman, identified as the wife of a BJP activist, threw the eggs targeting Patnaik in protest against the detention of her husband ahead of the chief minister's arrival in the district.
The ruling BJD, of which Patnaik is the president, condemned the incident.
"The act is condemnable. The opposition party (BJP) is intolerant of the success of Chief Minister Naveen Patnaik and is resorting to such hooliganism. There are many instances of opposition parties being intolerant," said Jeevan Pradip Das, BJD MLA from the district.
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