The decision was made after the Jerusalem District Court ruled his extradition was possible.
The family of Naama Issachar appealed to the court to stop the extradition in the hopes that Burkov could be returned to Russia as part of a deal that will secure the release of the young Israeli woman from a Russian prison where she is serving a seven and a half year prison sentence for drug charges.
Lawyers for the Issachar family say they will no appeal the decision to the Supreme Court.
The minister's decision followed consultations held by Israeli officials including Prime Minister Benjamin Netanyahu, in light of the request of both Russia and the United States that the hacker be extradited to them, however the American request came first and the Supreme court ruled in their favor.
Alexei Burkov was arrested in Israel in December 2015 on suspicion he had committed cyber fraud in the United States.
Two weeks ago, he was moved to a more secure wing in prison.
The American Secret Service launched a criminal investigation into an internet site called Cardplanet.cc after they suspected the platform sold credit and debit card information belonging mostly to U.S. citizens or issued by American banks.
The investigation revealed Burkov managed the site since its inception and until 2013.
In August 2015, a federal grand jury in the eastern district of Virginian indicted him and issued a warrant for his arrest.