Western leaders have congratulated Dmitry Medvedev on becoming Russia's new president, but election observers have highlighted flaws in the poll.
Mr Putin (left) congratulated Mr Medvedev late on Sunday
European Commission President Jose Manuel Barroso said he was confident that the EU-Russia "strategic partnership" would develop.
German congratulations were mixed with regret about apparent irregularities in Mr Medvedev's landslide poll win.
In Moscow, police arrested dozens of people ahead of an opposition rally.
More than 100 officers swooped on opposition activists as they were gathering for the unauthorised demonstration in the Russian capital on Monday.
Some of the protesters lit flares and chanted: "Your election is a farce!"
An authorised march in Mr Putin's home city of St Petersburg drew 2,000 demonstrators, including ex-chess champion and Other Russia opposition leader Garry Kasparov.
Mr Medvedev, President Vladimir Putin's chosen successor, won 70.23% of the vote on Sunday, with nearly all votes counted, according to official results.
"I think (my presidency) will be a direct continuation," he said, referring to Mr Putin's eight years in office.
His nearest rival was Communist leader Gennady Zyuganov, with 17.76% of the vote. Mr Zyuganov vowed to go to court over alleged fraud, Itar-Tass news agency reports.
Britain said it would "judge the new [Russian] government on its actions", while French Foreign Minister Bernard Kouchner said the EU and Russia had to start a new dialogue.
In Washington, a White House spokesman said "the United States looks forward to working with him [Medvedev]" - but he avoided commenting on the election itself.
Earlier, the head of observers from the Parliamentary Assembly of the Council of Europe (PACE), Andreas Gross, said Russia's "democratic potential" was unfulfilled.
Riot police dispersed an unauthorised protest in Moscow
Mr Gross, whose team was the only Western observer mission in Russia, said candidates had been denied equal access to the media, and registration procedures made it hard for independents to stand.
But he conceded that Mr Medvedev had a solid mandate.
The PACE 22-member observer mission said the elections "had more the character of a plebiscite on the last eight years in this country".
It added: "The president-elect will have a solid mandate given to him by the majority of Russians."
Mr Putin was barred by the constitution from seeking a third consecutive term but is expected to become prime minister, after his United Russia party won a huge majority in December's parliamentary election.
Last month, the Organisation for Security and Co-operation in Europe (OSCE), the continent's main election watchdog, decided not to monitor the presidential election because of Moscow's restrictions.
HAVE YOUR SAY
From the West's standpoint, Medvedev is by far a more attractive leader to deal with
Yevgeni, Moscow, Russia
Only 300 international observers were monitoring the 96,000 polling stations on Sunday.
After the polls, Mr Medvedev said he would control foreign policy and make the defence of Russian interests his chief priority. He confirmed that Mr Putin would be moving out of the Kremlin.
The mainly pro-Kremlin media have welcomed Mr Medvedev's victory.
Izvestia newspaper said the vote was a "triumph of the majority" that created a "bridge to the future for Russia".