US Attorney General Eric Holder said the extradition was a “victory for law and order” saying Mr Bout had been “a source of concern around the world”.
Russia described the extradition as a “blatant injustice”, the result of extreme American pressure on Thailand.
Mr Bout, who denies all charges, will appear at a US court on Wednesday.
The former Russian air force officer, 43, has been accused of trying to sell arms to Colombian rebels and supplying weapons that fuelled conflicts in Africa and the Middle East.
It has now lost that battle and has reacted furiously… But its strong words may be driven by much more than a simple desire to help Mr Bout as a Russian citizen.
Well-placed sources in Moscow have told the BBC Mr Bout could not have operated what is alleged to have been a massive illegal arms dealing business without support from many different parts of the Russian state… in particular the secret services.
Many suspect Mr Bout is himself an intelligence officer or at least closely connected with the intelligence agencies here.
A former British minister who had access to secret files while in office has alleged that Mr Bout’s connections go to very senior levels of the Russian government.
If these allegations are correct, then Mr Bout’s extradition to the US will be very worrying because highly sensitive information could be handed over to the Americans.
Following a long legal battle, Thailand eventually backed the US request for his extradition.
He was taken to Bangkok airport on Tuesday in handcuffs, escorted by armed police officers and with snipers deployed along the route.
If convicted, he faces a maximum penalty of life in prison.
But Mr Bout denies being, or ever having been, an arms dealer – and Moscow also insists he is innocent.
“Viktor Bout has been indicted in the United States, but his alleged arms trafficking activity and support of armed conflicts in Africa has been a cause of concern around the world,” said US Attorney General Holder in a statement.
“His extradition is a victory for the rule of law worldwide.”
Mr Bout is thought to have knowledge of Russia’s military and intelligence operations, and Russian diplomats fear the revelations he might make in open court, correspondents say.
Russian Foreign Minister Sergei Lavrov reiterated Russia’s belief that the US pursuit of Mr Bout was politically motivated, and said Russia would use all legal means to support him.
“Contrary to two rulings by a Thai criminal court which concluded that Viktor Bout’s guilt was not proven, he has still – by a decision of the Thai government – been extradited to the United States,” Mr Lavrov told Rossiya TV on Tuesday.
“I consider this to be unprecedented political pressure on the judicial process and on the government of Thailand. This whole story is an example of blatant injustice. We, as a state, will continue to render all necessary assistance to Viktor Bout as a Russian citizen.”
A Russian embassy official told the BBC that the Russian consul had also been unable to see Mr Bout.
Dubbed the Merchant of Death by a British politician, he was alleged to have supplied arms to Angola, Liberia and the Democratic Republic of Congo.
But he is also suspected of having used his network of air freight companies to supply weapons, in the early 1990s, to Afghanistan and Bosnia.
A website that describes itself as “The official site of Viktor A Bout” says he is a businessman with an undying love for aviation and an eternal drive to succeed.
The website says he started his career in the army of the former Soviet Union – and it was when the Soviet Union collapsed that he started buying up surplus Antonov and Ilyushin cargo planes.