Champagne corks popped on Friday in Moscow as Russians celebrated the start of Donald Trump's presidency, confident of better relations ahead between the two countries.

"It's weird, but it's great, and for the first time ever Russians are applauding the victory of a US presidential candidate. It's a sign of the times," political analyst Stanislav Byshok said.

Trump's promises to fix ravaged relations with Moscow have elated Russia's political elite following spiraling tensions with Washington over the Ukrainian crisis, the war in Syria and allegations of Russian meddling in the US election.

"We are ready to do our share of the work in order to improve the relationship," Prime Minister Dmitry Medvedev said on Facebook.

About 100 Trump sympathisers, nationalists and spin doctors gathered at a hipster party just a few hundred metres away from the Kremlin to celebrate Friday, with a triptych of Trump, Russian President Vladimir Putin and French nationalist politician Marine Le Pen in the centre of the hall.

An hour before Trump took the stage in Washington, the sound of opening champagne bottles echoed in the vaulted hall. The party was co-sponsored by the conservative Tsargrad TV channel, which is led by ultra-right ideologue Alexander Dugin.

"Yes, it's a holiday," said a beaming Dmitry Rode, a communications executive with a glass of champagne in his hand.

"We all hope that relations between our countries and more importantly between our peoples will help to develop our economies. We're neighbours, we're just 50 kilometres (30 miles) away from each other."

Some party-goers wore Guy Fawkes masks, associated with hackers, in a sly reference to charges that Russia interfered in the US election.

Trump's praise for Putin has raised expectations that he could move to normalise ties, although Trump hasn't articulated a clear policy and some of his Cabinet nominees have made hawkish statements on Russia.

Leonid Slutsky, the head of the foreign affairs committee in the lower house of parliament, expressed hope that Trump will move to establish constructive ties with Moscow, but cautioned that there is no "magic button" to instantly achieve that.

"We expect a slow but steady revival of our relations," he said.

Dmitri Trenin, director of the Carnegie Moscow Centre, predicted that Moscow will face a pragmatic but very tough partner in Trump.

"Russia's potential is incomparable to that of the United States," he said, adding that Moscow will have to apply a lot of skills "to play from the position of weakness and not lose."

Speaking at the World Economic Forum in Davos, Switzerland, Russian Deputy Prime Minister Igor Shuvalov voiced hope that Trump will work with Putin on solving the Ukrainian crisis and other problems, but warned against expectations of quick progress.

"Difficulties will remain," he said.