In Damascus, the Syrian presidency said in a statement that Assad had agreed to the reduction in the Russian air force presence, and denied suggestions that this reflected a difference between the two countries
"The whole subject happened in complete coordination between the Russian and Syrian sides, and is a step that was carefully and accurately studied for some time", the statement said, adding that Moscow had promised to continue support for Syria in "confronting terrorism".
Syria regards all rebel groups fighting Assad as terrorists.
Rebels and opposition officials alike reacted sceptically.
"I don't understand the Russian announcement, it's a surprise, like the way they entered the war. God protect us," Fadi Ahmad, spokesman for the First Coastal Division, a Free Syria Army group fighting in the northwest, said.
Opposition spokesman Salim al-Muslat demanded a total Russian withdrawal. "Nobody knows what is in Putin's mind, but the point is he has no right to be in be our country in the first place. Just go," he said.
A European diplomat was also sceptical. "It has the potential to put a lot of pressure on Assad and the timing fits that," the diplomat said.
"However, I say potentially because we've seen before with Russia that what's promised isn't always what happens."
MOMENT OF TRUTH
The Geneva talks are the first in more than two years and come amid a marked reduction in fighting after last month's "cessation of hostilities", sponsored by Washington and Moscow and accepted by Assad's government and many of his foes.
Russia's U.N. ambassador Vitaly Churkin confirmed some forces would stay in Syria. "Our military presence will continue to be there, it will be directed mostly at making sure that the ceasefire, the cessation of hostilities, is maintained," he told reporters at the United Nations in New York.
However, he added: "Our diplomacy has received marching orders to intensify our efforts to achieve a political settlement in Syria."
Speaking before Putin's announcement, de Mistura said Syria faced a moment of truth, as he opened talks to end a war which has displaced half the population, sent refugees streaming into Europe and turned Syria into a battlefield for foreign forces and jihadis.
The limited truce, which excludes the powerful Islamic State and Nusra Front groups, is fragile. The warring sides have accused each other of multiple violations and they arrived in Geneva with what look like irreconcilable agendas.
The Syrian opposition says the talks must focus on setting up a transitional governing body with full executive power, and that Assad must leave power at the start of the transition. Damascus says Assad's opponents are deluded if they think they will take power at the negotiating table.
In a sign of how wide the rift is, de Mistura is meeting the two sides separately, at least initially. The talks must focus on political transition, which is the "mother of all issues", the U.N. envoy said.