"My daughter Kimia, you have triggered the happiness of all the Iranians, and particularly of the women. I wish you eternal joy," the president tweeted.
The message is accompanied by a picture showing Alizadeh, 18, draped in the Iranian flag celebrating her Thursday victory in Rio, where she clinched bronze by beating Nikita Glasnostic of Sweden 5-1 in the taekwondo under-57kg division.
Even conservatives voiced their satisfaction over the teenager's win.
Fars news agency, which is close to the deeply conservative camp in Iran, hailed "Kimia who made history" and said in a report that the bronze medal she won "is worth gold".
"What an honour... to be the first," wrote Fars, describing Kimia as the "lionness of Iran" and adding that "one must be a woman to totally feel this moment with pride".
In keeping with Iran's strict Muslim custom, the teenager competed wearing a head scarf over with her taekwondo uniform and protective gear.
Foreign Ministry spokesman Hossein Jaberi Ansari congratulated Alizadeh as well Hedaya Malak of Egypt, who landed the other women's bronze in taekwondo, in a statement on his Instagram page.
He said the presence on the podium of the two veiled women was the "symbol of the unity and the efforts of Muslim women, who shine and yet respect their values".
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Alizadeh's victory was also celebrated on social networks by many of her compatriots, including popular actress Taraneh Alidoosti who sparked debate in Iran this year after she was pictured with a "woman power" symbol tattooed on her arm.
"The future will tell what you have achieved for your peers," tweeted Alidoosti.
"You have bolstered their faith and showed them that (sports) belongs to them as well," she wrote.
Iranian women are banned from entering stadiums for major sporting events, including football, as part of a male-female segregation ushered in by Iran's 1979 Islamic revolution.
In 1992, Lida Fariman was the first Iranian woman to compete at the Olympics in Barcelona in shooting.
This year, the Iran team at the Olympics comprised nine women out of a total of 41 athletes.
Alizadeh told Iranian television she was very excited about her victory in Rio and hopes Iranian women will be able to clinch gold at the next Olympics.
Her father was likewise elated.
"I hope that the Iranian people appreciate what Kimia has achieved," Keyvan Alizadeh told Mehr news agency. "She gave herself 100 percent".
He said an injury had prevented the young woman from grabbing the gold.