"The Arab and Muslim people... Who have always seen the Islamic republic as a supporter of the oppressed... And every fighter who resists for God, consider you today a beacon of hope," the group said.
Rowhani, a cleric viewed as a moderate political figure, was declared the winner of Iran's presidential election yesterday.
Iran is Hezbollah's biggest backer and supporter, and has also thrown its weight behind the regime of Syrian President Bashar al-Assad, helping it battle an uprising now in its third year.
Earlier this year, Hezbollah chief Hassan Nasrallah said Syria's "friends," including Iran, would not allow the Syrian regime of Assad to fall.