The Communist Party is expected to come in a distant second place, with the pro-Kremlin Liberal Democrats and Fair Russia also set to win seats in some of the assemblies.
About 3,600 separate elections took place across Russia on Sunday, involving around 20 million eligible voters out of Russia's population of 142 million people.
Figures on mayoral and district council votes were not immediately available.
Gennady Zyuganov and Vladimir Zhirinovsky, respective leaders of the Communist Party and the Liberal Democratic Party, said their votes were down because of manipulation.
They said their representatives were prevented from meeting voters and handing out party information.
Ivan Melnikov, the Communist party's electoral headquarters chief, said there had been a "return to shady technologies of the 1990s".
"We noted illegal campaigning, attempts to pressure voters, and the results are determined not by appealing to the voter, but by throwing in ballots through the early voting mechanisms," he was quoted as saying by RIA Novosti.
United Russia, which was formed by the Kremlin during Putin's 2000-2008 presidency, said it was the opposition which had violated election rule.
It said voters had been bribed with gifts including cheap wine and chocolate.
Putin and Dmitry Medvedev, the Russian president, are not involved directly in the elections, and the nation's two biggest cities - Moscow and St Petersburg - did not hold votes.