Saleh met his counterpart Hassan Rouhani to discuss trade, transport links and efforts to tackle dust pollution that plagues areas on both sides of the border.
"It is now time for a new order in the region that would meet the interests of all countries in it," Saleh told a Tehran news conference, according to a translation provided by Iran's state broadcaster.
"We believe Iran has an important role to play in this new order," he added.
Saleh, a 58-year-old moderate Kurd, was elected last month to the largely ceremonial role of president.
Rouhani said the two countries aimed to boost trade from $12 billion to $20 billion.
"Security and stability in the whole region is to everyone's benefit. We do not need any foreign interference in the region," Rouhani said.
Iran has been highly influential in Iraq since the toppling of Saddam Hussein in 2003, and was a key partner in pushing back the Islamic State group in recent years.
But Iran has also been blamed for factionalism and poor governance in Iraq and its consulate in the southern city of Basra was burnt to the ground during violent protests earlier this year.
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