AFPTV images from Damascus showed rebels firing into the air at sunrise, with some flashing the victory sign and crying "Allahu akbar", or God is greatest.
Some people climbed atop a tank in celebration, while others battered a toppled statue of Assad's father, Hafez.
"I can't believe I'm living this moment," tearful Damascus resident Amer Batha told AFP by phone.
"We've been waiting a long time for this day," he said, adding: "We are starting a new history for Syria."
Assad's reported departure comes less than two weeks after the Islamist Hayat Tahrir al-Sham (HTS) group challenged more than five decades of Assad family rule with a lightning offensive.
"After 50 years of oppression under Baath rule, and 13 years of crimes and tyranny and (forced) displacement... we announce today the end of this dark period and the start of a new era for Syria," the rebel factions said on Telegram.
Prime Minister Mohammed al-Jalali said he was ready to cooperate with "any leadership chosen by the Syrian people".
The head of war monitor the Syrian Observatory for Human Rights, Rami Abdel Rahman, told AFP: "Assad left Syria via Damascus international airport before the army security forces left" the facility.
AFP was unable to immediately confirm the report.
HTS said their fighters broke into a jail on the outskirts of Damascus, announcing an "end of the era of tyranny in the prison of Sednaya", which has become a by-word for the darkest abuses of Assad's era.The rapid developments came just hours after HTS said it had captured the strategic city of Homs.
Syria's defence ministry earlier denied that rebels had entered Homs, describing the situation there as "safe and stable".
Homs is about 140 kilometres (85 miles) from the capital and was the third major city seized by the rebels, who began their advance on November 27, reigniting a years-long war that had become largely dormant.
US President Joe Biden was keeping a close eye on the "extraordinary events" unfolding in Syria, the White House said.
The Britain-based Observatory confirmed "the doors of the infamous 'Sednaya' prison... have been opened for thousands of detainees who were imprisoned by the security apparatus throughout the regime's rule".
Assad's government had previously denied the army had withdrawn from areas around Damascus.
His rule has for years been backed by Lebanese group Hezbollah, whose forces "vacated their positions around Damascus", a source close to it said early Sunday.
Reports that Assad had fled were followed by the premier saying he was ready to "cooperate" with a new leadership and any handover."This country can be a normal country that builds good relations with its neighbours and the world... but this issue is up to any leadership chosen by the Syrian people," Jalali said in a speech broadcast on his Facebook account.
Rebel factions aired a statement on Syrian state television, saying they had toppled the "tyrant" Assad and urged fighters and citizens to safeguard the "property of the free Syrian state".
State TV later broadcast a message proclaiming the "victory of the great Syrian revolution".
AFP has been unable to independently verify some of the information provided by the government and the rebels.
Before the announcements, Damascus residents had described to AFP a state of panic as traffic jams clogged the city centre, with people seeking supplies and queueing to withdraw money.
But Sunday morning saw chants and cheering, with celebratory gunfire and shouts of "Syria is ours and not the Assad family's".
At the dawn call to prayer, some mosques broadcast religious chants usually reserved for festive occasions, while also urging residents to stay at home.
Before the rebels entered Damascus, the Islamist-led alliance had wrested control of Aleppo, Hama and also Homs, known during the early years of the civil war as the "capital of the revolution".The Observatory said Daraa, cradle of the 2011 uprising, also fell from government control.
The commander of Syria's US-backed, Kurdish-led Syrian Democratic Forces (SDF), which controls much of northeast Syria, hailed "historic" moments with the fall of Assad's "authoritarian regime".
Jordan had urged its citizens to leave neighbouring Syria "as soon as possible", as have the United States and Assad ally Russia, which both keep troops in Syria.
An Iraqi security source said Saturday Baghdad had allowed in hundreds of Syrian soldiers who "fled the front lines" through the Al-Qaim border crossing.
Iraq on Sunday evacuated its embassy staff to Lebanon, a diplomatic source said.
Assad ally Iran's embassy was attacked "by unknown individuals", Iranian state media reported, but its staff had already left the facility, according to other reports citing the foreign ministry.
HTS is rooted in the Syrian branch of Al-Qaeda. Proscribed as a terrorist organisation by Western governments, it has sought to soften its image in recent years, and told minority groups living in areas they now control not to worry.
Since the offensive began, at least 826 people, mostly combatants but also including 111 civilians, have been killed, the Observatory said.
The United Nations said the violence has displaced 370,000 people.
Describing Syria's civil war years as a "dark chapter", UN special envoy to Syria Geir Pedersen said in a statement Sunday that "today we look forward with cautious hope to the opening of a new one -- one of peace, reconciliation, dignity, and inclusion".
UAE presidential adviser Anwar Gargash declined to confirm or deny speculation that Assad had sought shelter in his country, but said he hoped Syrians would "work together" to avoid "another episode of impending chaos".
US President-elect Donald Trump posted on his Truth Social platform that "Assad is gone", adding Russia "was not interested in protecting him any longer".
For years Assad was propped up by Russia and Iran, while Turkey, which historically backed the opposition, on Sunday urged a "smooth transition" in Syria.
China said Sunday it was following the situation, and hoped Syria "returns to stability as soon as possible".
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