The woman suffered burns on one side of her face and shoulders, but is currently out of danger, police said.
This is the latest in a series of attacks reported this year by foreign tourists in India.
In January, five men were arrested for allegedly kidnapping and raping a Japanese tourist in the eastern Indian city of Kolkata (Calcutta).
And in September, an American tourist said she was raped by two men in the northern city of Dharamsala.
Varanasi police superintendent Akash Kulharni told BBC Hindi that officers looking into the acid attack suspected the involvement of the Russian tourist`s landlord`s son, who disappeared after the attack.
They are waiting for the victim to regain consciousness so that they can record her statement, police added.
According to the official statistics, there are a few hundred acid attacks in India every year.
There has been legislation to restrict the sale of acid in the country, but campaigning groups say it is still easily available.
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