This is demonstrably false. Far-right parties failed to pass a 5% barrier to enter parliament, although if they had banded together, and not split their vote, they would have probably slipped past the threshold.
Only one government minister has links to nationalist parties - though he is in no way a neo-Nazi or fascist. And the speaker of parliament, Volodymyr Groysman, is Jewish. He has the third most powerful position in the country after the president and prime minister.
But Ukrainian officials and many in the media err to the other extreme. They claim that Ukrainian politics are completely fascist-free. This, too, is plain wrong.
As a result, the question of the presence of the far-right in Ukraine remains a highly sensitive issue, one which top officials and the media shy away from. No-one wants to provide fuel to the Russian propaganda machine.
But this blanket denial also has its dangers, since it allows the ultra-nationalists to fly under the radar. Many Ukrainians are unaware that they exist, or even what a neo-Nazi or fascist actually is, or what they stand for.
This hyper-sensitivity and stonewalling were on full display after President Petro Poroshenko presented a Ukrainian passport to someone who, according to human rights activists, is a "Belarusian neo-Nazi".
The Ukrainian leader handed out medals on 5 December to fighters who had tenaciously defended the main airport in the eastern region of Donetsk from being taken over by Russian-backed separatists.
Among the recipients was Serhiy Korotkykh, a Belarusian national, to whom Mr Poroshenko awarded Ukrainian citizenship, praising his "courageous and selfless service".
The president's website showed a photo of Mr Poroshenko patting the shoulder of the Belarusian, who was clad in military fatigues.
Experts who follow the far right have strongly objected to President Poroshenko's decision.
They say Mr Korotkykh was a member of the far-right Russian National Unity party and also a founding member of the neo-Nazi National Socialist Society (NSS) in Russia.
According to Ukrainian academic Anton Shekhovtsov, the NSS's main goal "is to prepare for a race war".
Mr Shekhovtsov said the Belarusian had been charged for involvement in a bombing in central Moscow in 2007, and was detained in 2013 in the Belarusian capital Minsk for allegedly stabbing an anti-fascist activist. He was later released for lack of evidence.
Even though the details involved accusations rather than facts, if true they were damning, said human rights activist Halya Coynash.
Top Ukrainian officials then rejected as defamatory any claims that Mr Korotkykh had neo-Nazi ties.
"Counter-intelligence has no information that could prevent him from receiving Ukrainian citizenship," said Valentyn Nalyvaychenko, the head of Ukraine's security services.
Nevertheless, the fact is, neo-Nazis are indeed a fixture in Ukraine's new political landscape, albeit in small numbers.
As Mr Korotkykh's case demonstrates, the ultra-nationalists have proven to be effective and dedicated fighters in the brutal war in the east against Russian-backed separatists and Russian forces, whose numbers also include a large contingent from Russia's far right.
As a result, they have achieved a level of acceptance, even though most Ukrainians are unfamiliar with their actual beliefs.
The volunteer Azov Battalion is a case in point.
Run by the extremist Patriot of Ukraine organisation, which considers Jews and other minorities "sub-human" and calls for a white, Christian crusade against them, it sports three Nazi symbols on its insignia: a modified Wolf's Hook, a black sun (or "Hakensonne") and the title Black Corps, which was used by the Waffen SS.
Azov is just one of more than 50 volunteer groups fighting in the east, the vast majority of which are not extremist, yet it seems to enjoy special backing from some top officials:
Ukraine's media have been noticeably silent on this subject.
Recently, prominent newspaper and online publication Left Bank published an extensive interview with Mr Troyan, in which the journalists asked no questions at all about his neo-Nazi past or political views.
And after the Unian news agency reported the presidential ceremony under the headline, "Poroshenko awarded Belarusian neo-Nazi with Ukrainian passport", it was soon replaced with an article that air-brushed out the accusations of extremism.
Unian's editors have declined to comment on the two pieces.
There are significant risks to this silence. Experts say the Azov Battalion, which has been widely reported on in the West, has damaged Ukraine's image and bolsters Russia's information campaign.
And although Ukraine is emphatically not run by fascists, far-right extremists seem to be making inroads by other means, as in the country's police department.
Ukraine's public is grossly under-informed about this. The question is, why doesn't anyone want to tell them?
Ever since Ukraine's February revolution, the Kremlin has characterised the new leaders in Kiev as a "fascist junta" made up of neo-Nazis and anti-Semites, set on persecuting, if not eradicating, the Russian-speaking population.