Iran’s Islamic Revolution Guards Corps (IRGC) said it shot down the drone over Iranian airspace, near Kuhmobarak in the southern province of Hormozgan.
IRGC commander Maj-Gen Hossein Salami said the incident sent “a clear message to America”, state media said.
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.
The US has 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.
Tensions were further fuelled on Monday when Iran said its stockpile of low-enriched uranium would next week exceed limits it agreed with world powers under a landmark nuclear deal in 2015.
Iran stepped up its production in response to tightening economic sanctions from the US, which unilaterally withdrew from the deal last year.
The IRGC said it shot down a US surveillance drone in the early hours of Thursday after it violated Iranian airspace, the official Irna news agency reported.
The drone was identified by the IRGC as a RQ-4 Global Hawk, but a US military official told Reuters news agency that the drone was a US Navy MQ-4C Triton.
The secretary of Iran’s Supreme National Security council warned on Wednesday that Iran would not permit any violation of its airspace.
“Our airspace is our red line and Iran has always responded and will continue to respond strongly to any country that violates our airspace,” Ali Shamkhani was quoted as saying by the semi-official Tasnim news agency.
Last week, the US military accused Iran of attempting to shoot down a US MQ-9 Reaper drone with a surface-to-air missile in an attempt to disrupt surveillance of one of the tankers attacked in the Gulf of Oman, the Kokuka Courageous.
The drone had earlier observed a fire on board the other vessel, the Front Altair.
The previous week, another US MQ-9 Reaper was shot down over Yemen by a surface-to-air missile fired by the Iran-backed rebel Houthi movement.
The US military said the altitude of the engagement “indicated an improvement over previous Houthi capability, which we assess was enabled by Iranian assistance”. Iran denies providing weapons to the Houthis.