The strikes reportedly hit rebel-controlled areas of Homs and Hama provinces, causing casualties.
The US says it was informed an hour before they took place.
Russian defence officials say aircraft targeted the Islamic State group, but an unnamed US official told Reuters that so far they did not appear to be targeting IS-held territory.
Syria’s civil war has raged for four years, with an array of armed groups fighting to overthrow the government.
The US and its allies have insisted that President Assad should leave office, while Russia has backed its ally remaining in power.
Russia’s decision to intervene with its air power greatly complicates the Syrian crisis while probably offering little additional chance of a diplomatic resolution.
Russian sources indicate that Sukhoi Su-24 warplanes were involved, operating out of an airbase near Latakia.
There are serious questions about who exactly the Russian aircraft are targeting. US officials believe that the initial Russian strikes are not in IS-held territory, raising the possibility that Russian air power is being utilised more in the form of close air support for Syrian government forces against the multiple enemies of the Assad regime.
Of course, many of these enemies are supported by the West’s Arab allies or Turkey. The warning time given by the Russians to the Americans announcing the start of their operations may also raise some eyebrows, suggesting that much moredetailed co-ordination may be needed in future to avoid incidents in Syrian airspace.
The upper house of the Russian parliament granted President Vladimir Putin permission to deploy the Russian air force in Syria.
The Russian defence ministry said the country’s air force had targeted IS military equipment, communication facilities, arms depots, ammunition and fuel supplies.
Syrian opposition activists said Russian warplanes had hit towns including Zafaraneh, Rastan ands Talbiseh, resulting in the deaths of 36 people, a number of them children.
None of the areas targeted were controlled by IS, activists said.
In a televised address, Mr Putin said the air strikes were targeting Islamist militants – including Russian citizens – who have taken over large parts of Syria and Iraq.
“If they [militants] succeed in Syria, they will return to their home country, and they will come to Russia, too,” he said.
He added that Russia was not going to send ground troops to Syria, and that its role in Syrian army operations would be limited.
“We certainly are not going to plunge head-on into this conflict… we will be supporting the Syrian army purely in its legitimate fight with terrorist groups.”
Mr Putin also said he expected President Assad to talk with the Syrian opposition about a political settlement, but clarified that he was referring to what he described as “healthy” opposition groups.
A US defence official said: “A Russian official in Baghdad this morning informed US embassy personnel that Russian military aircraft would begin flying anti-Isil [IS] missions today over Syria. He further requested that US aircraft avoid Syrian airspace during these missions.”
US state department spokesman John Kirby told reporters: “The US-led coalition will continue to fly missions over Iraq and Syria as planned and in support of our international mission to degrade and destroy Isil [IS].”
More than 250,000 Syrians have been killed and a million injured in four-and-a-half years of armed conflict, which began with anti-government protests before escalating into a full-scale civil war.
More than 11 million others have been forced from their homes, four million of them abroad, as forces loyal to President Assad and those opposed to his rule battle each other – as well as jihadist militants from IS and other groups. Growing numbers of refugees are going to Europe.
Regional and world powers have also been drawn into the conflict. Iran and Russia, along with Lebanon’s Hezbollah movement, are propping up the Alawite-led government. Turkey, Saudi Arabia and Qatar are backing the Sunni-dominated opposition, along with the US, UK and France.