At its peak, the storm brought winds of up to 70mph, knocking over trees and damaging windows.
Such strong storms are unusual in Tehran, correspondents say.
The storm caused power cuts and traffic accidents from poor visibility as dust and sand engulfed parts of Tehran.
Some domestic flights to central Iran have been diverted, AP news agency reports.
One shopkeeper in Tehran, identified only as Nouri, told AP news agency: “It was a horrifying storm and suddenly everywhere went dark.
“I closed my store’s shutters to prevent damage. [A] big tree broke and knocked in the windows, shattering them.”
The total number of deaths is unclear. Iranian state television said five were killed, while state-run Irna news agency said four died.
Some of the deaths were reportedly caused by falling trees.
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