Haider al-Abadi still faces opposition closer to home, where his Shia party colleague Nuri al-Maliki has refused to step aside after eight years as premier.
Other neighbors Turkey and Saudi Arabia also welcomed Abadi's appointment.
A statement from Maliki's office said he met senior security officials and army and police commanders to urge them "not to interfere in the political crisis".
Underscoring the convergence of interest in Iraq that marks the normally hostile relationship between Washington and Iran, senior Iranian officials congratulated Abadi on his nomination, three months after a parliamentary election left Maliki's bloc as the biggest in the legislature. Like Western powers, Iran is alarmed by ISIL militants' hold in Syria and Iraq.
"Iran supports the legal process that has taken its course with respect to choosing Iraq's new prime minister," Ali Shamkhani, the secretary of the Supreme National Security Council was quoted as saying.
"Iran favors a cohesive, integrated and secure Iraq," he said, adding an apparent appeal to Maliki to concede.
Iraqi state television said Abadi "called on all political powers who believe in the constitution and democracy to unite efforts and close ranks to respond to Iraq’s great challenges".
One politician close to Abadi told Reuters that the prime minister-designate had begun contacting leaders of major groups to sound them out on forming a new cabinet. The president said on Monday he hoped he would succeed within the next month.
Kurdish president Masoud Barzani told US Vice President Joe Biden that he would work with Abadi, the White House said.
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