The billionaire, also the target of European Union sanctions, was described by the UK government as part of Russian president Vladimir Putin’s inner circle.
Chelsea have been operating under the terms of a strict government licence that expires on May 31, meaning they have been unable to renew contracts for existing players and have faced a transfer ban.
Completion of the takeover has been a lengthy process due to concerns over the potential for Abramovich to profit from the sale of the London club.
But the Premier League gave the green light on Tuesday (24) evening and hours later ministers said they would permit Chelsea’s sale.
Sale will not benefit Roman Abramovich or other sanctioned individuals: Culture secretary Nadine Dorries
Culture secretary Nadine Dorries, whose brief includes sport, tweeted: “Given the sanctions we placed on those linked to Putin and the bloody invasion of Ukraine, the long-term future of the club can only be secured under a new owner.
“We are satisfied the proceeds of the sale will not benefit Roman Abramovich or other sanctioned individuals.”
A consortium led by Boehly, a co-owner of baseball’s Los Angeles Dodgers, beat off competition to agree to a record deal to buy the club earlier this month.
At one stage, it seemed close to collapse over concerns proceeds would not reach good causes as promised by outgoing owner Abramovich.
The Russian, who bought Chelsea in 2003, denied he had asked for his £1.5 billion loan to the club be repaid when the sale was concluded.
The government said in its statement it was satisfied that the “full proceeds” of the sale would not benefit Abramovich or any other sanctioned individual.
“We will now begin the process of ensuring the proceeds of the sale are used for humanitarian causes in Ukraine, supporting victims of the war,” it added.
Of the total investment, £2.5 billion will be deposited into a frozen UK bank account to be donated to a charitable foundation for the benefit of victims of the war in Ukraine.
The new owners will commit £1.75 billion in further investment “for the benefit of the club”.
Former Chelsea chief executive Peter Kenyon told the BBC that the end of the saga would be a “complete relief” for everyone connected to the club.
“We’ve got a very short transfer window and plans will have been made before,” he said.
“They have already lost some players through not being able to re-sign them so I think everybody – the squad and the management – will be relieved they can get on with the job they’re employed to do.”
Chelsea have been transformed since Abramovich bought them in 2003 for just £140 million, winning 19 major trophies in his 19-year reign — including five Premier League titles and two Champions League triumphs.