Other opposition supporters braved temperatures of minus 45C to attend a protest in Yakutsk.
"Vladimir Putin has once again thrown Alexei Navalny in jail because he is afraid of his own people-and what they would do if they were free", Cotton said in a statement.
President Putin's most vocal critic is barred from standing in the election.
Navalny warned that authorities planned to clamp down on his youngest supporters, tweeting a screenshot of a text message sent around ahead of the rallies.
Police also detained several members of Navalny's team.
Elsewhere, there were 600 demonstrators in Russia's third-most populous city, Novosibirsk, and 550 protesters in Nizhny Novgorod in western Russian Federation, the ministry said.
"In order to take down our broadcast, the police cut out the door to the Anti-Corruption Foundation (FBK) office, and then began to saw the door to the studio right in the middle of broadcast", he said in a Facebook post. "You didn't come out for me, but for your future", he added.
Six of Mr Navalny's supporters at the Moscow studio were detained, as well as around 16 protesters in other parts of Russian Federation, independent monitoring group OVD-info said. Several protesters said they were there first and foremost to oppose the government rather than to support Navalny.
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Earlier in the day, the Russian Police raided Navalny's Moscow office, questioning supporters and shutting off his online television broadcast and detaining some officials.
Putin won the previous election in 2012 on a turnout of 65 percent and authorities are pulling out all the stops to boost the figures this year.
However, the main protests in Moscow and St. Petersburg were not agreed with the authorities. "There is nobody to vote for", the 27-year-old said.
"We announce a boycott of the election".
Navalny himself served three jail sentences of 15 days, 25 days and 20 days for organising unauthorised protests a year ago.
In power as president or prime minister for nearly two decades, Putin is almost certain to win another six-year term as president March 18.
Mr Putin, who refuses to mention Mr Navalny by name, retains a massive approval rating in Russian Federation and is widely expected to win a fourth six-year term in office.
"I want change", Andrei Petrov, 20, told AFP in the former imperial capital.
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