The group of monuments at Pattadakal in Bagalkot and Gol Gumbaz in Vijayapura will remain open for visitors from sunrise to 9 pm.
At present, the gates of all monuments under the protection of the Archaeological Survey of India (ASI) are closed by 5.30 pm.
The Union culture ministry decided to extend the visiting hours at these two ASI-protected monuments in Karnataka along with eight others in various states, accepting the demands of stakeholders.
The ministry had received requests for extending the visiting hours at a total of 52 ASI-protected monuments.
"We are doing this as an experiment. If it succeeds, we will consider extending the visiting hours at other ASI-protected monuments also," Culture Minister Prahlad Singh Patel said on Monday. The decision comes into effect "immediately", he said.
The other monuments which will now remain open for visitors till 9 pm are Humayun's Tomb and Safdarjung's Tomb in Delhi; Markanda Group of Temples in Gadchiroli, Maharashtra; Sheikh Chilli's Tomb in Thanesar near Kurukshetra in Haryana; Duladeo Temple in Khajuraho, Madhya Pradesh; Rajarani Temple Complex in Bhubaneswar, Odisha; Man Mahal, Vaidhshala in Varanasi; and Rani-ki-Vav in Patan, Gujarat.
The visiting hours at ASI-protected monuments in different parts of the country have been kept between 9 am and 5.30 pm for various reasons, including the safety and security of tourists.
"The situation has changed now. While the ASI has its own security arrangements at most of the monuments, the tourist spots also have adequate lighting arrangements and basic amenities," Prahlad Singh Patel said.
To ensure tourists' safety, the minister said, local administrations have been asked to provide adequate security arrangements at all the 10 monuments where visiting hours have been extended.