"What I would hope, is that Rouhani... Use that (new) term to begin a process of dismantling Iran's network of terrorism, dismantling its financing of the terrorist network, dismantling the manning and the logistics and everything that they provide to these destabilising forces that exist in this region," Tillerson said in Riyadh.
"We also hope that he puts an end to their ballistic missile testing," Tillerson said at a joint press conference with his Saudi counterpart, on the first day of a visit by President Donald Trump to Saudi Arabia.
Rouhani won 57 per cent of the vote, scoring a convincing victory over hardline cleric Ebrahim Raisi on 38.3 per cent.
Jubeir said the elections in the Islamic republic were an "internal Iranian matter", in an attempt not to comment.
"Who they choose for their president is their business, as it should be. From our perspective we judge Iran by its actions, not by its words," he added.
Shiite-dominated Iran is the arch rival of Sunni heavyweight Saudi Arabia, which along with other Gulf monarchies accuse Tehran of meddling in their internal affairs.
"Our people have declared to neighbouring countries and the whole region that the path to ensuring security is the reinforcement of democracy and not relying on foreign powers," he said.
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