In the video, which has been retweeted on the Iranian president's unofficial Twitter account, Rouhani is shown along with traditional musical instruments, banned on state media since the Islamic revolution of 1979.
A voice-over commentary attached to the video highlighted the significance of the retweet.
"The fact that the president's personal Internet site decided to pick up this video shows that it is a major step forward in breaking taboos and supporting artists," the commentary said.
It surfaced amid jubilation among his moderate and reformist supporters at the hopes of rapprochement with the West raised by a nuclear deal struck with major powers on Sunday.
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It shows Iran's supreme leader and bastion of political orthodoxy Ayatollah Ali Khamenei, to whom Rouhani addressed the August speech.
It also includes clips of revolutionary leader Ayatollah Ruhollah Khomeini.
But it has segments in all of Iran's main minority languages -- Arabic, Azeri, Baluchi and Kurdish -- as well sign language, in a clear bid to rally the entire country behind the Rouhani government's drive to mend fences with the outside world.