Medvedev conveyed this in a congratulatory letter to his S. Sudanese counterpart Salva Kiir Mayardit.
Early Saturday, South Sudan became the 193rd country recognised by the UN and the 54th UN member state in Africa.
Oil-rich Sudan, Africa's largest country, has served as a 20th-century battleground for land, resources and ethnic identity.
South Sudan, where most follow Christian and traditional beliefs, became independent after a January referendum agreed under a 2005 peace deal that ended decades of civil war with the Arab Muslim north.
"The fact that South Sudan has obtained national sovereignty has opened a gate to enhanced dialogue, based on principles of mutual respect, friendship, and mutually beneficial cooperation, permeated by successful projects," Medvedev's letter said.
Russian businessmen are looking for joint projects with colleagues from South Sudan, especially in such areas as extraction of commercial minerals, energy and agriculture, the Russian president continued.
"I expect that our partnership will be one of the vital factors, enhancing stability on the African continent."
A range of countries, including Russia, the US, China, Switzerland and others have already formally recognised the Republic of South Sudan.
More than 30 African leaders and senior world politicians gathered Saturday in South Sudan's capital - Juba - to celebrate its first day as a nation.
US President Barack Obama, while granting his country's recognition of South Sudan as an independent state, said: "I am proud to declare that the United States formally recognizes the Republic of South Sudan as a sovereign and independent state upon this day, July 9, 2011."
"Today is a reminder that after the darkness of war, the light of a new dawn is possible," he added.