Several people were injured at the Sina Athar clinic, and there are fears the death toll will rise further.
Iran's state media quoted senior officials as saying the blast was caused by a gas leak.
Television pictures showed plumes of smoke billowing in the sky. Firefighters extinguished the blaze several hours later.
Most of the victims of Tuesday's blast were women, according to the authorities.
Some of the victims "were in upper floors, in operation rooms - who were either patients being operated on or those with them", Tehran fire department spokesman Jalal Maleki was quoted as saying by state media.
"They unfortunately lost their lives due to the heat and thick smoke," he said, adding that firefighters had rescued 20 people.
The clinic, which carries out light surgeries and medical imaging, had 25 employees inside at the time of the explosion, Mr Maleki said.
The incident comes several days after a big explosion near a military facility rocked Tehran.
The cause of that incident was also said by the authorities to have been a gas leak.
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