Mr Lavrov's comments came a day after US Vice-President Joe Biden visited the Ukrainian capital and gave promises of support for the pro-Western government and a warning to Russia not to interfere in Ukraine.
The crisis in Ukraine, in its fourth month, has dragged Russia's relations with the West to their lowest point since the Cold War.
In the east, pro-Russian armed separatists have seized about a dozen public buildings and are defying Kiev's authority. A further escalation could lead to damaging economic sanctions and raises the risk of a disruption to the Russian gas supplies on which Europe depends.
Nato says Russia has built up a force of about 40,000 troop on its border with Ukraine. Moscow says some are stationed permanently, while others have been deployed as a precaution to protect Russia from the instability in Ukraine.
In Moscow, Mr Lavrov said Russia would respond if its interests, or the interests of its citizens, were attacked.
"Russian citizens being attacked is an attack against the Russian Federation," he said in an interview with the Russia Today news channel. Referring to developments in Kiev, he added: "There is no reason not to believe that the Americans are running the show."
Russia justified its intervention in Crimea earlier this year by saying it had to defend Russians living there. Some people in eastern Ukraine hold Russian passports.
Mr Lavrov's ministry, in a separate statement, accused the US and the interim government in Kiev of a "distorted interpretation" of an international accord, signed in Geneva last week, under which illegal armed groups in Ukraine are to disarm and give up buildings they have occupied.
Russia said that criterion applies not only to the pro-Russian separatists in the east, but also to groups in the Ukrainian capital whose protests helped bring Ukraine's new government to power.
"Instead of taking effective measures to implement the ... agreements, Kiev, Washington and a series of European capitals continue to insist that it is only Ukrainian citizens defending their rights in the south-east of Ukraine who need to give up their weapons," a ministry statement said. Earlier, Ukraine's government relaunched a security operation to crack down on the pro-Russian armed groups after an Easter pause and said it had the backing of the United States. But it was unclear what steps Kiev could take to restore authority in the mainly Russian-speaking east without wrecking the Geneva deal.
"The security forces are working on the liquidation of illegal armed groups," in the east of Ukraine, First Deputy Prime Minister Vitaly Yarema told reporters.
"The corresponding activities will be carried out in the near future, and you will see the results."
The Interior Ministry said it had flushed armed separatists out of a town they had controlled in eastern Ukraine in an "anti-terrorism" operation. It said the operation took place on the outskirts of the town of Sviatogorsk and that no-one was injured. There had been no previous reports of gunmen in the town, which lies just outside the stronghold of pro-Russian militants in Slaviansk.
Kiev's decision to resume its operation in the east was prompted in part by the discovery of two bodies in eastern Ukraine. One was that of politician Volodymyr Rybak, a member of the same party as Ukraine's acting president.
The Ukrainian government appeared emboldened by Mr Biden's visit on Tuesday. He urged Russia to curb separatist militias in the east.