The report quoted the Guard's navy chief, General Ali Fadavi, as saying that the "intruding" drone was captured, but gave no further details on the location or when the incident occurred.
However a US Navy spokesman said no American drones were missing in the Middle East. He also cast doubt on Iranian claims that the US ScanEagle drone entered Iranian airspace, saying US operations in the Persian Gulf are "confined to internationally recognised water and airspace."
Other nations in the Gulf, including the United Arab Emirates, have ScanEagle drones in service.
Tehran later said it recovered data from the RQ-170 Sentinel, a top-secret drone equipped with stealth technology. Gen Fadavi said the unmanned Scan Eagle aircraft was now in Iran's possession.
"The US drone, which was conducting a reconnaissance flight and gathering data over the Persian Gulf in the past few days, was captured by the Guard's navy air defence unit as soon as it entered Iranian airspace," he said. "Such drones usually take off from large warships."
There was no immediate comment from the US Navy's 5th Fleet, which is based in Bahrain. If true, the seizure of the drone would be the third reported incident involving Iran and US drones in the past two years.
Last month, Iran claimed that a US drone had violated its airspace. The Pentagon said the unmanned aircraft came under fire - at least twice but was not hit - and that the Predator was over international waters.
In 2011, Iran claimed it brought down a CIA spy drone after it entered Iranian airspace from its eastern borders with Afghanistan and Pakistan. The RQ-170 Sentinel drone, which is equipped with stealth technology, was captured almost intact. Tehran later said it recovered data from the top-secret drone.