Iran's Islamic Revolution Guards Corps (IRGC) said the aircraft had violated Iranian airspace, and that the incident sent a "clear message to America".
But the US military insisted the drone had been over international waters at the time, and condemned what it called an "unprovoked attack" by the IRGC.
It comes at a time of escalating tension between the US and Iran.
On Monday, the US defence department said it was deploying 1,000 extra troops to the region in response to "hostile behaviour" by Iranian forces. It has already sent an aircraft carrier strike group and B-52 bombers.
The US has also accused Iran of attacking two oil tankers with mines last Thursday just outside the Strait of Hormuz, in the Gulf of Oman. Iran rejects the allegation.
It was the second time in a month tankers had been attacked close in the region, through which a fifth of the world's oil passes each day.
Tensions were further fuelled on Monday when Iran announced its stockpile of low-enriched uranium would next week exceed limits it agreed with world powers under a landmark nuclear deal in 2015.