In a statement issued here, Khamenei said the "cruelest people on earth" assassinated the "honourable" commander who "courageously fought for year against the evils and bandits of the world", reports the Tehran-based Press TV.
"His demise will not stop his mission, but the criminals who have the blood of General Soleimani and other martyrs of the Thursday night attack on their hands must await a vigorous revenge.
"... The continued fight and achievement of the final victory will make life bitterer for the murderers and criminals," he added.
Besides offering condolences to Soleimani's family, Khamenei also declared three days of mourning.
The IRGC said that Soleimani and Abu Mahdi al-Muhandis, the second-in-command of the Hashd Shaabi or the Iraqi Popular Mobilization Forces (PMF), were killed in the US strike that targeted their vehicle on the Baghdad International Airport road.
Iranian Foreign Minister Mohammad Javad Zarif called the US attack as "extremely dangerous and a foolish escalation".
"The US' act of international terrorism, targeting and assassinating General Soleimani, the most effective force fighting IS, Al Nusrah, Al Qaeda, is extremely dangerous and a foolish escalation.
"The US bears responsibility for all consequences of its rogue adventurism," he said in a series of tweets.
Meanwhile, the PMF has also confirmed the incident, adding that its public relations director Mohammed Reza al-Jaberi and four other members were also killed after three Katyusha rockets struck a military base next to the Baghdad airport earlier on Friday.
It described the attack as a "cowardly US bombing".
President Trump have not officially commented on the development, but he posted a picture of the American flag on his Twitter handle.
Since 1998, Maj Gen Soleimani led Iran's Quds Force - the IRGC's elite unit which handles clandestine operations abroad.
( With inputs from IANS )