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The reformist-backed cleric won just over 50% of the vote and so avoided the need for a run-off.
Mr Rouhani said his win was a "victory of moderation over extremism".
Thousands of Iranians have taken to the streets of Tehran, shouting pro-reform slogans and hailing Hassan Rouhani's election as president.
One of Mr Rouhani's main election pledges was to try to ease international sanctions imposed on Iran over its nuclear programme.
The US expressed concern at a "lack of transparency" and "censorship" but praised the Iranian people and said it was ready to work with Tehran.