Several thousand supporters of Russian opposition leader Alexei Navalny rallied in central Moscow as he disputed the results of mayoral polls in which he narrowly failed to push a Kremlin ally into a run-off, claiming fraud.

Around 10,000 people gathered for a peaceful rally in the city centre after the Moscow election commission released the final tally showing that an ally of President Vladimir Putin, incumbent mayor Sergei Sobyanin, had scraped through in the first round with 51.3 percent of the vote.

In a major surprise, Navalny, a charismatic 37-year-old leader of the opposition movement who campaigned under the shadow of a controversial conviction for embezzlement, polled far more strongly than projected with over 27.2 percent.

"At these elections, politics was finally born in Russia. The opposition was born. We know for sure what to do, we know for sure how to do it," he told a cheering crowd.

Yet Navalny contended that the results were falsified and urged the authorities to hold a recount that he said would lead to a second-round runoff.

The rally took place on a central Moscow square, which was the focal point of huge anti-Putin protests in the winter of 2011-12 that saw Navalny become the star speaker.

Dmitry Yagodkin, 42, said he turned up for the rally because he believed the authorities had manipulated the results.

"I believe that our votes have been stolen, a second round is essential," he said.

"A second round would be a victory," said Alexander Koidan, a 21-year-old law student.

Detective novelist Boris Akunin called on Sobyanin to come out to the protesters and explain himself.

"Like many people, I have a lot of suspicions about the results of the Moscow elections," he said at the rally, interviewed by TV Rain.

The Moscow authorities had given Navalny permission to hold a rally of no more than 2,500 people.

On Monday evening, police said more than 9,000 people had gathered and Navalny's campaign said they had been fined for breaching the limit.

Navalny warned supporters not to stage any illegal protests following the rally after the prosecutor-general's office gave him an official warning over his calls for "civil disobedience".

"I don't plan to let you down or put you in danger," he said.

-- 'Unsubstantiated claims' --

In a nationwide day of local polls whose results may worry the Kremlin, opposition anti-drugs campaigner Yevgeny Roizman defeated a pro-Kremlin candidate in elections in Russia's fourth-largest city of Yekaterinburg.

Putin congratulated all the winners in the polls, pointedly noting that the campaign was over and urging everyone to begin "joint positive work".

The chief of the Moscow election commission dismissed Navalny's claims as "pure PR".

"There are no facts apart from unsubstantiated claims about wholesale violations that are not backed by anything," Valentin Gorbunov told journalists.

Nevertheless, Sobyanin said he was ready to meet Navalny for talks.

Analysts said that Navalny's stronger than expected showing had made him a force to be reckoned with.

"From a civil activist he has turned into a politician. Twenty-seven percent is a colossal trump card," said Maria Lipman, an analyst with the Carnegie Moscow Center.

"As a result of the poll he has become even more dangerous" (for the Kremlin), she said.

Before the election, which analysts saw as a crucial test of the protest mood in Russia more than a year into Putin's new Kremlin term, nearly all pollsters had forecast Navalny would receive 20 percent of the vote.

Many said they voted for Navalny even if some might have reservations about his tough anti-immigrant rhetoric because he pushed for political change.

Navalny's campaign insists that Sobyanin had polled less than 50 percent.

Voter turnout in the mayoral race stood at a meagre 32 percent, which appeared to have helped Navalny, with the protest leader far more successful in bringing out core support than Sobyanin's low-key push for votes.

--- 'Don't rock the boat' ---

The candidacy of the charismatic anti-corruption crusader Navalny made the race the first genuinely competitive Russian election since the turbulent early post-Soviet years.

It was also the first time in a decade that the Kremlin had allowed Muscovites to elect their mayor, and Sobyanin clearly wanted to pick up popular legitimacy after being appointed in 2010 to replace longstanding mayor Yury Luzhkov.

In a late-night Sunday rally in central Moscow attended by thousands and lit up by fireworks, Sobyanin said he was sure of victory and congratulated himself for organising "the most honest and open elections in the history of Moscow".

In an apparent dig at Navalny, he called on his rivals not to "rock the boat" and concede defeat.

In July, Navalny was sentenced to five years in a penal colony on fraud charges that he says were trumped up. He was jailed, but suddenly released a day later pending his appeal.

At Monday's rally he said he expected the authorities would give him a "real or suspended jail sentence. They will drag me through the courts."