Security fears were already high during this hajj season because of sectarian strife between Sunni and Shia Muslims in Iraq and elsewhere in the region.
"We have not changed our plans. We have prepared for the worst, but so far things have been very quiet," said a security officer at Mena outside Mecca, who asked not to be named.
Many Sunni Arabs on hajj were angered by the news; but some Iraqi pilgrims said Saddam's death was a gift on the occasion of Eid al-Adha, the Feast of Sacrifice that began on Saturday.
Nadir Abdullah, an Iraqi pilgrim, said: "Congratulations, this is like two Eids! I hope God will not have mercy on him."
Eid al-Adha, the most important day in the Islamic calendar, marks the beginning of the three-day stoning ritual at the Jamarat Bridge when pilgrims symbolically cast out sins.
With such large crowds, Saudi Arabia deploys more than 50,000 security men to try to avoid deadly stampedes, as well as attacks by Islamists opposed to the US-allied Saudi royals.
Hundreds of thousands of elated pilgrims, clad in white robes symbolising equality and selflessness, managed to move over the Jamarat Bridge on Saturday without incident.
In the worst hajj tragedy in 16 years, 362 pilgrims were crushed to death in January as crowds surged across the bridge to throw stones at three large walls representing the spot where Islam says Satan tempted the biblical patriarch Abraham.
New construction work
"Circulation has been fluid, people have managed to get on and off the bridge with ease," said Zohra Qanqum from Morocco.
New construction work allows 250,000 pilgrims to pass over the bridge in safety each hour.
For the first time, Saudi authorities have also removed most of the pilgrim squatters who in past years have set up tents at the side of the bridge.
"This is incomparable to previous years. I managed to throw my stones in less than an hour early in the morning," said Mohammed Abdel-Jabbar from Sudan.
Saudi government-backed clerics have encouraged pilgrims to stone throughout the day, challenging those who insist on the afternoon and thus risk a surge of pilgrims at one time.
Pilgrims, mostly moving on foot, cover a 44km circuit around Mecca during the gruelling five-day ritual.
They must repeat the stoning twice more on Sunday and Monday. Saudi officials said on Saturday that 2.4 million pilgrims were performing hajj.
But with hundreds of thousands of local residents sneaking into Mecca without official permits, the total number could be closer to three million.