"No organisation or institution has the right to carry out such military operations inside the territory of our country without permission and agreement from the Islamic Government of Afghanistan," a statement from his office said.
"Afghanistan condemns this act by NATO and announces that such unilateral operations are a clear violation of Afghan sovereignty as well as international law, and any repetition will be met by the required reaction from our side."
The statement said Karzai had ordered the ministries of defence and interior to investigate the operation, which took place late Thursday in the eastern province of Nangahar, and to report back to him by Saturday night.
A Russian official however said Karzai was "misinformed" in saying the operation went ahead without permission from Afghan authorities.
The representative in Kabul of the Russian anti-drugs service, Alexey Milovanov, told AFP it was an Afghan operation.
"It was an operation conducted by the Afghan ministry of interior, not by us," said Milovanov.
"We have simply acted as advisers, according to an agreement between our two countries permitting the presence of Russian advisers during a drug raid."
He said that four Russian anti-drug officials were present at the operation.
Russian media on Friday quoted the head of Russia's anti-drugs service, Viktor Ivanov, as saying that Afghan officials had been involved in the raid.
Russia frequently criticises what it describes as the inadequate anti-drug policies of United States and NATO forces in Afghanistan, which it says lead to an increased flow of drugs into Russia through Central Asia.
Ivanov travelled to Washington last week to discuss co-operation in fighting drug trafficking, and accused the United States of failing to destroy heroin laboratories and crack down on poppy-growing landowners.
Russian drug control authorities have estimated that 30,000 Russians died in 2009 from using Afghan heroin, and that a million have died in the past decade.
The apparent disagreement comes as NATO is trying to encourage greater Russian engagement in Afghanistan, more than 20 years after the former Soviet Union withdrew forces after a decade-long war against mujahideen insurgents.
NATO chief Anders Fogh Rasmussen will visit Russia next week for talks to lay the groundwork for a major Russia-NATO summit, Moscow's NATO ambassador told the ITAR-TASS news agency on Saturday.
"We are expecting the secretary general to arrive on the evening of November 2 but the whole working part of the visit will be on November 3," Russian NATO envoy Dmitry Rogozin told the agency, speaking in Brussels.
NATO said earlier that Rasmussen planned to meet President Dmitry Medvedev to see if the two sides can "move forward" on a NATO request for Moscow to provide around 20 helicopters to Afghan forces.
Russia already assists in the training of counter-narcotics officials outside Moscow to combat the Afghan heroin trade.
"There will be work to see if we can enhance our cooperation in training counter-narcotics officials," Appathurai added.
Afghanistan produces about 90 percent of the world's opium in an industry estimated to be worth almost three billion dollars a year, which helps fund the Taliban-led insurgency.
Afghanistan produced an estimated 3,600 tonnes of opium this year, almost 50 percent of the 2009 output, the UN said in a recent report, but added that the value of the opium rose by 38 percent at the farm gate.
A report by the UN Office on Drugs and Crime put the value of opium output at five percent of Afghanistan's gross domestic product (GDP) this year, and said this was more than six times the value of the country's wheat crop.
The Taliban, who have been waging war for almost nine years, are believed to get much of their funding from Afghanistan's drug production.
Their presence in parts of the south, particularly the central Helmand valley where much of Afghanistan's poppy is grown, is directly linked to cultivation and distribution of opium and heroin.
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