Vasyunyk, quoted by local news agencies after a cabinet meeting, said the agency headed by a former cabinet minister would be replaced by a 50-strong "coordinating bureau" responsible to the government.
Ukraine, which is co-hosting the championship with Poland, has repeatedly come under fire for being slow to tackle the mammoth tasks required to modernise stadiums, hotels and transport networks.
UEFA President Michel Platini has made two visits to Kiev this year and UEFA warned the two countries in September that their role as hosts could be in jeopardy if they did not make "the necessary efforts".
The president of Ukraine's soccer federation, Grigory Surkis, told Ukrainian media: "UEFA has consistently stressed that it is not happy with the agency's work."
In Warsaw, Polish Sports Minister Miroslaw Drzewiecki said he was unconcerned by what he saw as a technical shake-up.
"I am not worried by this at all. The decision is just an adjustment to the requirements and expectations of UEFA," Drzewiecki told TVP Info news channel.
"Their wish was for Ukraine to implement a similar model to the one we have in Poland."
Senior Ukrainian, Polish and UEFA officials were upbeat last week about preparations for Euro 2012 and said the two ex-communist neighbours would be successful hosts.
The cabinet meeting also endorsed technical and financial specifications for the renovation of Kiev's main stadium at an estimated cost of more than two billion hryvnias ($340 million).
Both countries have had problems building or renovating stadiums -- Kiev lost a year through legal wrangling over an adjacent shopping centre and then switched general designers.
