Hasan Rouhani's comments came as Syrian troops and rebels fought some of the fiercest battles in the mountains of the coastal province of Latakia, an Assad stronghold.
Rouhani made the comments during a meeting in the Iranian capital of Tehran yesterday with Syrian Prime Minister Wael al-Halqi, Syria's state news agency SANA said.
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Iran has been one of Assad's staunchest backers since Syria's crisis began. Tehran is believed to have supplied Assad's government with billions of dollars since the country's crisis began in March 2011.
Iran-supported Hezbollah also has sent fighters into Syria to bolster an offensive by Assad forces.
"The Islamic Republic of Iran aims to strengthen its relations with Syria and will stand by it in facing all challenges," SANA quoted Rouhani as saying in a report from Tehran.
"The deep, strategic and historic relations between the people of Syria and Iran ... Will not be shaken by any force in the world."
Rouhani was elected in June and was endorsed by the country's supreme leader on Saturday, allowing him to begin acting as president. He was sworn in Sunday.
Assad vowed last evening to crush the rebels trying to overthrow him, saying Syria is between two choices: "a state of the law or a state run by thieves and bandits."
Assad spoke while taking part in an iftar, the meal that breaks the dawn-to-dusk fast during the Muslim holy month of Ramadan. The iftar was attended by government officials, religious leaders and members of the country's unions and political parties, SANA said.
It was Assad's second public appearance this week after he visited troops in the Damascus suburb of Daraya on Thursday to mark Army Day.
Assad said "terrorism cannot be dealt with politically but should be struck with an iron fist."
Assad's troops have achieved victories in the past two months mostly near the capital Damascus and in the strategic central province of Homs that links the capital Damascus with Assad's stongholds on the coast.
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