Images purporting to show an unnamed US warship were shown on television.
Iran's navy commander praised the operation for getting so close to a warship "to get such accurate footage of the combat units of foreign forces".
The US said an Iranian drone had flown over one of its carriers recently, but did not say if it was this incident.
It had not posed a risk to the USS Harry S Truman, but the move was "abnormal and unprofessional", Reuters news agency quoted a US navy spokeswoman as saying.
Nicole Schwegman said that incident had occurred on 12 January, which is the same day Iran briefly detained 10 US sailors who had entered Iranian territorial waters by mistake.
The report by Iran's state Irinn news channel did not say when the surveillance drone had made its flight except to say it was on the third day of a naval exercise.
The Associated Press news agency said this suggested it had happened on Friday, as Iran's naval drill began earlier this week.
An Iranian light submarine also closely participated in the surveillance operation, Iran's state-run Irna news agency reported.
Adm Habibollah Sayyari, Iran's navy chief, told state TV it was a sign of the "bravery, experience and the scientific capabilities of our drone operators".
International sanctions against Iran over its nuclear programme were lifted this month, somewhat easing its relations with the West.
But within a few days, the US had imposed new sanctions related to Iran's ballistic missile programme.
Iran said the sanctions, which prevent 11 entities and individuals linked to the programme from using the US banking system, had "no legal or moral legitimacy".

Adrian Brown made the "unscheduled landing" between Bassingbourn Primary School and the village college at about 08:40 GMT on Tuesday.
Unable to reach Royston Heath because of light winds, and not wanting to "disturb sheep" in a nearby field, he chose the playground.
Primary head Sue Brown said the unexpected visitor "made their day".
Mr Brown, from Litlington in Cambridgeshire, arrived on his single-person "hopper" balloon - one that does not have a basket -  "just as the bell was ringing", Mrs Brown said.
Read more on this and other stories from Cambridgeshire
He said he had set off 30 minutes earlier from a field north of Bassingbourn and was aware the wind conditions were light and variable.
"Ideally we like to land close to a road or track to enable easy access for the retrieve crew to come and collect us," Mr Brown said.
"With such light winds I didn't think we would be able to reach the heath at Royston.
"The Royston Road from Litlington has the complication of power lines and sheep in the fields which I did not want to disturb."
When he landed, Mr Brown was greeted by a familiar face - his daughter, who is a pupil at Bassingbourn Village College.
"[She] came across and said hello, which was lovely," Mr Brown said.
The engineering company boss has been ballooning for 25 years.
Iran flew an unarmed drone over a US aircraft carrier and took "precise" photographs during a naval drill in the Gulf, its state broadcaster said.
[NEXT_CONCEPT]
A man landed his hot air balloon on a Cambridgeshire school playing field "to avoid sheep", he said.