First Deputy Prime Minister Medvedev won just over 70% of votes on Sunday, and will be inaugurated as president, replacing Vladmir Putin, on May 7.
Chinese leader Hu Jintao spoke to Medvedev on the phone, congratulating him and wishing him success. Hu also invited him to visit China, an offer Medvedev accepted, a spokesman for the president-in-waiting said.
The spokesman also said French President Nicholas Sarkozy "warmly congratulated" the election winner and invited him to France. "The invitation was gratefully accepted," he added.
A White House spokesman issued a statement saying United States President George W. Bush was ready to work with Medvedev and would contact him later this week.
U.S. National Security spokesman Gordon Johndroe said: "The United States looks forward to working with him. It's in our mutual interest for Russia and the United States to work together on areas of common interest such as non-proliferation, counterterrorism and combating transnational crime."
European Commission President Jose Manuel Barroso also congratulated Medvedev. A spokesman for Barroso quoted him as saying the next Russia-EU summit to be held this summer would provide an opportunity for the EU leaders to meet the new Russian president.
British Prime Minister Gordon Brown sent a letter of congratulations to the future Russian president, but did not invite him to Downing Street.
The premier's spokesman said it was too early to determine the prospects of Anglo-Russian relations, and that the U.K. would judge the new Russian leadership on "actions and the results of those actions."
Relations between the two countries have been strained by issues including the murder of former Russian security officer Alexander Litvinenko in London in November 2006, and closure of British Council offices in regions earlier this year.
A spokesman for German Chancellor Angela Merkel quoted her as wishing Medvedev "good luck and success with the difficult task ahead."
However, spokesman Thomas Steg said: "It is without doubt the case that during the election campaign situations arose where it became clear that democratic rules were not always upheld. The government made it clear in the run-up to Sunday's vote that we regret that the international observers could not carry out their job as well as one would wish."
The election was boycotted by the main European election monitoring body, the Organization for Security and Cooperation in Europe (OSCE), in protest against restrictions imposed on observers.
The only group of Western observers to attend the polls was from the Parliamentary Assembly of the Council of Europe. The organization called the election doubtful, citing problems with candidate registration and equal media access for candidates. However, PACE conceded that Medvedev's victory reflected the will of Russian voters.
Japan's Chief Cabinet Secretary Nobutaka Machimura told a news conference that Tokyo hoped the new Russian president would work to solve a long-running territorial dispute with Japan.
"We hope the new president will come up with a solid solution to the Northern Territories [South Kurils] issue so that Japan-Russia relations will be rebuilt to reach an upgraded dimension," Kyodo news agency quoted him as saying.
Austrian Foreign Minister Ursula Plassnik welcomed Medvedev's election as president, saying he "so far has shown a clear understanding of interweaving and interdependence of international economic relations."
Italian Foreign Minister Massimo D'Alema said he had sent a congratulatory letter to Medvedev.
Greek Prime Minister Costas Karamanlis also congratulated the future president.
Serbian President Boris Tadic sent a telegram congratulating Medvedev and saying: "Serbia will always be a reliable friend and partner for Russia," his press service said.
Finnish President Tarja Halonen telephoned the future president to congratulate him, and invited him to visit Finland.