"We can only wipe out this evil through joint efforts," Dmitry Medvedev told Pratibha Patil.
Patil said she shared Medvedev's view, and that it was crucial for the international community to work together to eradicate terrorism.
Medvedev and Indian Prime Minister Manmohan Singh issued a joint statement earlier on Friday urging the international community to provide assistance to bring to justice the organizers of last week's terrorist attacks in Mumbai, which left over 170 people dead.
The statement called on all countries "to actively cooperate with the Indian authorities" to support efforts to find "the perpetrators, masterminds, sponsors...and everyone connected with these barbaric acts," the declaration adopted by the Indian prime minister and the Russian president said.
Terrorists swept through Mumbai on November 26, armed with submachine guns and grenades attacking hotels, the railway station, a cinema, and a hospital. The three-day rampage killed 174 people, including more than 20 foreigners.
The Indian government said the 10 terrorists, who travelled to Mumbai by sea from Karachi, had been trained in Pakistan. Investigators have said the only terrorist captured alive, 21-year-old Azam Amir Kasav, has confessed to receiving military training in Pakistan.
On Monday, India sent an official note to Pakistan's ambassador in New Delhi demanding the extradition of 20 people believed to be involved in terrorist activities, including three fugitives wanted in India.
India and Russia also called for the development of a single approach to countering terrorist threats which would also cover the security problems in Afghanistan.
Russia and India expressed concern about the deterioration in the security situation in Afghanistan which has resulted in intensified attacks by the Taliban across the country.
