Speaking in a Twitter clip, the president said: “What is happening today is the writing of a history written by every Syrian citizen.
“The writing did not start today, it started six years ago when the crisis and war started against Syria.”
He said Syria, the region and the international situation would be different after “liberating Aleppo”.
It comes as hundreds of civilians started to leave the east of the city, headed for rebel-held areas outside Aleppo as part of a ceasefire deal.
Images on Syrian television showed at least 17 buses and 10 ambulances in the first convoy.
The Red Cross said 3,000 civilians had so far left in two convoys and a third was heading to the city after dusk.
Some 50,000 people, mostly civilians, are still in eastern Aleppo, according to Staffan de Mistura, the UN’s peace envoy to Syria.
He said between 1,500 and 5,000 rebel fighters remained.
US Secretary of State John Kerry tore into President Assad on Thursday, saying he had carried out “nothing short of a massacre” and there was “no excuse” for the brutality aided by Russia and Iran.
He said the ceasefire appeared to be holding, but that there were reports of injured people being attacked.
A rescue service spokesman claimed ambulances had been shot at by pro-Assad militias, wounding three people.
Witnesses said the first buses appeared full as they left the Ramousah district, which has been under siege for months.
Russia said rebel fighters – whose territory in the city has all but been wiped out – would be transported towards Idlib.
Soldiers will escort them under orders from President Putin, according to state news agencies.
The Russian defence ministry said Syrian authorities had guaranteed their safety and added it would use drones to monitor the convoy.
In a tit-for-tat arrangement, another 29 trucks and ambulances headed to the villages of al Foua and Kefraya, which have been besieged by rebels.
Aid trucks accompanied by UN vehicles have also reportedly crossed into northern Syria to help people desperate for food, shelter and medicine.
The evacuation comes after a new agreement was reached after an initial ceasefire collapsed on Wednesday amid more airstrikes and shelling.
The Ahrar al Sham rebel group said negotiators had overcome obstructions by Iran and its militias.
A child waves from inside a bus as she is evacuated from rebel-held areas in Aleppo
Rebel fighters and their families are being evacuated under a ceasefire deal
A convoy of ambulances and buses, carrying nearly 1,000 people, left in the first phase
A staff member of the Syrian Red Crescent stands on a bus amid rebels fighters and their families
A wounded boy sits inside an ambulance
The rebel withdrawal will pave the way for President Assad’s forces to reclaim complete control of the city. Click through for more
The buses were heading for Idlib, west of Aleppo
Turkey is said to have played a major role in getting Russia to pressure President Assad to honour the agreement.
Despite the deal, anger still pervades among western governments.
Foreign Secretary Boris Johnson summoned the Russian and Iranian ambassadors to “convey in person the Government’s profound concern over the situation in Aleppo”.
He said they had failed to uphold international humanitarian law by not helping aid reach besieged residents.
“They deserve no credit for the fact that an evacuation appears to be underway today,” he added.
“Having inflicted such suffering on the people of eastern Aleppo, Iran and Russia cannot expect praise for allowing some people to escape at the final hour.”
Theresa May said the UK would give an extra £20m to help vulnerable people leaving the city.
“We can’t bring back those who are lost, but we can save those who remain,” the PM said.
The ceasefire comes after recent advances that drove rebels into a small pocket of the city. Hundreds of civilians have been killed and tens of thousands displaced.
Despite losing ground in Aleppo, the rebels still control large areas of Syria – as does Islamic State.