Khamenei accused Riyadh of "a blame game", after the Saudi foreign minister said Iran was playing politics with tragedy.
"Instead of passing the buck and playing a blame game, the Saudis should accept their responsibility and apologise to the world's Muslims and the bereaved families," Khamenei said in comments reported by Iran's official IRNA news agency.
Iranian leaders have been fiercely critical of Saudi authorities' handling of safety at the hajj, following Thursday's stampede during a ritual stoning of the devil in Mina, near the holy city of Mecca.
Iranian President Hassan Rouhani, addressing a UN development summit in New York yesterday, said he wanted to "emphasise the need for swift attention to the injured as well as investigating the causes of this incident and other similar incidents in this year's hajj."
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Iranian Attorney General Ebrahim Raeisi earlier called what happened "not only incompetence, but a crime".
Saudi Foreign Minister Adel al-Jubeir, also in New York for the UN General Assembly, said the Iranians "should know better than to play politics with a tragedy that has befallen people who were performing their most sacred religious duty."
It was the worst disaster to strike the annual pilgrimage in a quarter-century.
"We will reveal the facts when they emerge. And we will not hold anything back," Jubeir said, adding the kingdom has a long history of devoting "tremendous resources" to ensuring a successful pilgrimage.
"I would hope Iranian leaders would be more sensible and more thoughtful with regards to those who perished in this tragedy, and wait until we see the results of the investigation."
Iran has demanded that affected countries have a role in the stampede investigation, and on Friday Ayatollah Mohammad Emami Kashani, leading the main weekly prayers in Tehran, said the kingdom is "incapable" of organising the hajj.