Russia's Federal Security Service (FSB, ex-KGB) identified the man as Ryan C Fogle -- third secretary of the political section of Washington's embassy in Moscow -- and said he had been handed back to the embassy after his detention.
The foreign ministry said it was summoning US ambassador Michael McFaul on Wednesday for an explanation and slammed Washington for what it described as "provocative acts in the spirit of the Cold War".
Photographs published by state English language television RT showed a baseball-capped Fogle being held to the ground face down and having his hands put behind his back for the arrest.
He was then shown being questioned at the Federal Security Service while documents such as his passport and a stack of 500 euro notes along with some letters were displayed.
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The FSB footage also displayed supposed espionage equipment including wigs, a torch, compass and even a mundane atlas of Moscow as well as a somewhat old-fashioned mobile phone.
The statement added that "recently, the US intelligence service has made repeated attempts to recruit the staff of Russian law enforcement agencies and special services".
In a video that RT said was provided by the FSB, Fogle is seen sitting down as a man, presumably a Russian security officer, tells the suspect about his alleged crime.
He is then accused of offering USD 100,000 for espionage to a security service employee who is involved in counterinsurgency work in the Russian North Caucasus.
The incident comes amid a new downturn in Russian-US relations sparked by the Syrian crisis and concern in Washington over what it sees as President Vladimir Putin's crackdown on all dissent.
The last major spy row between the two former Cold War rivals involved the glamourous Anna Chapman and 10 other Russian spies arrested in the United States in 2010.
The spy scandal -- which ended with their swap for four Russians convicted of spying for the West -- was a huge embarrassment for Russia's foreign intelligence at the time.