He fell into a coma after drinking what his supporters suspect was poisoned tea; they accuse the authorities of trying to conceal a crime.
Doctors treating him in Omsk had insisted on Friday that he was too ill to be moved.
But they later said his condition was stable enough for the flight. His wife Yulia is travelling with him.
A medical evacuation flight, paid for by the German NGO Cinema for Peace, is taking Mr Navalny to Berlin, where he will be treated at the Charité hospital.
“Alexei’s plane has taken off for Berlin,” his spokeswoman Kira Yarmysh tweeted on Saturday morning. “Massive thanks to everyone for their support. The struggle for Alexei’s life and health is just beginning,”
Ms Yarmysh earlier said it was a pity doctors had taken so long to approve his flight as the plane and the right documents had been ready since Friday morning.
Mr Navalny fell ill during a flight from Tomsk to Moscow on Thursday, and his plane made an emergency landing in Omsk. A photograph on social media purported to show him drinking from a cup at a Tomsk airport cafe before the flight. His team suspects a poisonous substance was put in his tea.
Disturbing video appeared to show a stricken Mr Navalny howling in agony on the flight. Passenger Pavel Lebedev said he heard the activist “screaming in pain”.
What the doctors said on Friday
The head doctor at the hospital where Mr Navalny was being treated in Omsk, Alexander Murakhovsky, warned late on Friday they did not recommend flying, “but his wife insists on her husband being transferred to a German clinic”.
“The patient’s condition is stable,” deputy chief doctor Anatoly Kalinichenko was quoted as saying by the Interfax news agency.
Mr Navalny is in an induced coma, and his condition is reported to be stable.
“As we’re in possession of a request from relatives to permit him to be transported somewhere, we have now taken the decision that we do not object to his transfer to another in-patient facility.”
Doctors said earlier that no poison had been found in his body, suggesting his condition might be the result of a “metabolic disorder” caused by low blood sugar.
Health officials then indicated that traces of an industrial chemical had been found on his skin and hair. The local interior ministry told the Rapsi legal news agency that the chemical was usually included in polymers to improve their elasticity, but its concentration was impossible to establish.
The prominent critic of President Vladimir Putin has consistently exposed official corruption in Russia. He has served multiple jail terms.
What do Mr Navalny’s supporters say?
At a news conference in Berlin, Mr Navalny’s aide Leonid Volkov said at first doctors at the hospital had been helping to facilitate his transfer but abruptly stopped doing so.
“[It was] like something was switched off – like medicine mode off, cover-up operation mode on – and the doctors refused to co-operate any more, refused to give any information even to Alexei’s wife,” he said.
“The doctors who were helping to do the paperwork to make the transportation of Alexei to Charité possible started to say that he’s not any more transportable, he’s not any more stable, contradicting themselves.”
The Cinema for Peace Foundation was founded by activist and filmmaker Jaka Bizilj. In 2018, it arranged for the treatment of Pyotr Verzilov – an activist with Russian protest group Pussy Riot – who had symptoms of poisoning.
Mr Verzilov’s ex-wife, activist Nadya Tolokonnikova, told BBC News that Mr Navalny’s condition resembles the “poisoning” of her ex-husband.
“What German doctors told me after not finding poison in my ex-husband’s blood is that the poison disappears in three days. So the Russian doctors only let him go when they were sure there was no traces of poison left,” she explained.
She also expressed surprise about what has happened to Mr Navalny: “I thought Alexei was so powerful as a political figure that Mr Putin would not interfere.”
Mr Navalny’s wife Yulia wrote to President Putin asking him to allow her husband to be moved. She feared the Russian authorities were stalling so that evidence of any chemical substance would be lost.
Mr Putin’s spokesman Dmitry Peskov said on Thursday that the Kremlin would help move Mr Navalny abroad if necessary and wished him a “speedy recovery”. On Friday he said it was purely a medical decision.
Foreign leaders including Germany’s Angela Merkel and France’s Emmanuel Macron have expressed concern for Mr Navalny. In the US, Democratic presidential candidate Joe Biden has described the incident as “unacceptable” and vowed that, if elected, he would “stand up to autocrats like Putin”.
Source: BBC and News Agencies