Boehly agreed terms for the purchase of the European champions for £4.25b in early May.
Abramovich put the club up for sale back in March for alleged links to Russia president Vladimir Putin, who is leading an invasion of Ukraine.
Abramovich had his assets frozen by the UK government and Chelsea have been operating under a special license ever since.
The Premier League and UK government approved the sale last week.
A statement released by Chelsea on Monday said: "Under the terms of the agreement, Boehly and Clearlake will share joint control and equal governance of the club. Boehly will serve as chairman of the holding company.
"Boehly and Clearlake are committed to investing in key areas that will extend and enhance Chelsea's competitiveness, including the redevelopment of Stamford Bridge, further investment in the academy, the women's team, and Kingsmeadow Stadium. The owners will also continue the important work of the Chelsea Foundation."
Upon confirmation of the takeover, Boehly told the club's website: "We are honoured to become the new custodians of Chelsea Football Club. We're all in, 100 per cent. Every minute of every match.
"Our vision as owners is clear: we want to make the fans proud. Along with our commitment to developing the youth squad and acquiring the best talent, our plan of action is to invest in the club for the long-term and build on Chelsea's remarkable history of success. I personally want to thank ministers and officials in the British government, and the Premier League, for all their work in making this happen."
Abramovich took over Chelsea in the summer of 2003 and injected over £1bn into the club, winning every major club honour available in that time including five Premier League titles, two UEFA Champions Leagues and two Europa Leagues.
