Kerry said Assad's regime must also provide "immediate and unfettered" access to inspectors from the Organisation for the Prohibition of Chemical Weapons (OPCW).
"The inspectors must be on the ground no later than November... and the goal is to establish the removal by halfway through next year," said Kerry, flanked by Lavrov.
The pressure is now on Assad to deliver, with Obama warning that "the United States remains prepared to act" if Damascus fails to comply.
"While we have made important progress, much more work remains to be done," Obama said.
Echoing Obama's warning, Kerry said there must be "no games, no room for avoidance of anything less than full compliance by the Assad regime".
But while Britain, France and the OPCW welcomed the deal, the rebels fighting Assad's regime greeted it with dismay, fearing it has scuppered any chance of Western intervention on their side.
"We cannot accept any part of this initiative," General Selim Idriss, the head of the Free Syrian Army, told reporters in Istanbul.
"Are we Syrians supposed to wait until mid-2014, to continue being killed every day, and to accept (the deal) just because the chemical arms will be destroyed in 2014."
And Iran, which has been one of Assad's main allies, said the United States no longer has a pretext to attack Syria.
Amid an intensifying diplomatic drive over the Syrian conflict, Kerry flies to Israel on Sunday to brief Prime Minister Benjamin Netanyahu about the deal, as well as discuss the Israeli-Palestinian peace process.
He will then travel to Paris for a Monday meeting with French counterpart Laurent Fabius and British Foreign Secretary William Hague as well as the Saudi Foreign Minister Saud al-Faisal.
Kerry said the steps agreed on Saturday would be encapsulated in a UN Security Council resolution drawn up under Chapter Seven of the organisation's charter, which provides for enforcement through sanctions including the possible use of military force.
But with Russia strongly opposed to the use of military threats against its long-term ally, and wielding a veto on the Security Council, Kerry acknowledged it was "impossible to have a pre-agreement" on what would happen in the event of non-compliance.
Lavrov signalled that Moscow would back some form of sanction, saying the Security Council would act under Chapter Seven if Syria fails to meet its demands.
Kerry said that Syria's bloody civil war could only be ended through negotiations, and he reiterated that he would meet with Lavrov again soon, this time in New York, to try to breathe life into planned peace talks between the regime and the opposition.
For Lavrov, Saturday's accord was an "excellent" agreement "whose significance is hard to overestimate".
Meanwhile, fighting on the ground in Syria continued unabated with rebel and regime forces engaged in a fierce battle for control of the ancient Christian town of Maalula, near Damascus.
Washington and Moscow hope to revive plans for peace talks in Geneva that seek a political transition to end the conflict that began in March 2011 with peaceful protests against the Assad regime and quickly turned violent following a brutal government crackdown of the demonstrations.
Russia's surprise announcement that Syria could hand over its chemical arsenal prompted Obama to put on hold military strikes the United States and France had threatened to unleash in response to an August chemical attack near Damascus, which Washington blames on the regime and says killed about 1,400 people.
The United States and Russia now agree that Syria possesses around 1,000 metric tonnes of various chemical agents, including mustard and sarin gas, sulphur and VX.
US officials also said there were around 45 sites that inspectors would have to check and Kerry said it would be feasible to do that, despite the fighting.
"One of the reasons we believe this is achievable is because the Assad regime has taken extraordinary means to keep control of these weapons," he added, noting that the chemical weapons were mainly in regions under Damascus' control.
EU foreign policy chief Catherine Ashton offered the bloc's help with "securing sites and in dismantling and destroying certain chemical agents".
NATO in turn welcomed the agreement, saying that "full and unreserved Syrian compliance is now key".
Despite the upbeat tone of Saturday's press conference, Kerry acknowledged that "a hard road" lies ahead amid widespread scepticism about Assad's good faith.
UN chief Ban Ki-moon has accused Assad of multiple crimes against humanity and said that a UN inspectors' report due to be published on Monday would provide "overwhelming" confirmation that chemical weapons were used on August 21.