The blast was triggered by a gas leak, Tehran Deputy Governor Hamid Reza Goudarzi told state television.
State TV showed the moment of an explosion after the initial blast started a huge blaze, as well as emergency services responding to the incident.
Ten women and three men died in the explosion, Deputy Health Minister Iraj Harirchi said on the TV.
A fire touched off by the blast has been extinguished, Jalal Maleki, the Tehran Fire Department spokesman said on state TV.
The clinic, which had 25 employees inside at the time of the blast, primarily carries out minor surgery and medical imaging, Harirchi said.
Last week, an explosion occurred close to a sensitive military site near Tehran which the defence ministry said was caused by a leaking tank at a gas storage facility. No deaths or injuries were reported.