Completion marks the end of the Abramovich era, which brought 21 trophies to Stamford Bridge in 19 years of ownership.
The Blues' future was thrown into doubt after sanctions were placed on now-former owner Abramovich by the UK government in March amid Russia's invasion of Ukraine.
Abramovich, the Russian-Israeli billionaire who bought Chelsea in 2003, was allowed to sell the club as long as none of the proceeds went to him, while the Blues' operations were limited under a special license.
It is understood the consortium led by Boehly, the LA Dodgers co-owner, and Clearlake Capital have acquired the club for £4.25billion - a record sum for a Premier League club - which is expected to restore normality in west London.
The UK government have recieved assurances the funds will go to support victims of the war in Ukraine, while around £1.5billion worth of debt was written off by Abramovich.
The consortium also includes Hansjorg Wyss, founder of the Wyss Foundation, and Mark Walter, co-founder and CEO of Guggenheim Capital.
After a complex and lengthy process, Boehly will share joint control with Clearlake Capital, although the American billionaire will be chairman of the holding company, making him the face of Chelsea's new era.
The 48-year-old promised to build on the club's success with big investment after 19 'unforgettable' years.
"We are honoured to become the new custodians of Chelsea Football Club," said Boehly.
"We're all in - 100% - 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."
And the UK government confirmed the club's long-term future has been secured by the takeover.
A government spokesperson said: "Today's change of ownership marks a new chapter for Chelsea Football Club in the best interests of its fans, the club and the wider football community.
"The club is now no longer subject to the sanctions imposed on Roman Abramovich, an individual who has enabled Putin's brutal and barbaric invasion of Ukraine.
"Since sanctions were imposed on Abramovich on 10 March, we have worked tirelessly to ensure the club can continue to play football, while maintaining the integrity of our sanctions regime.
"Chelsea's long-term future is now secured and binding commitments have been received which ensure sanctioned individuals cannot financially benefit from the sale. The Government retains control to ensure that this is the case.
"We have begun the process of ensuring the proceeds are used for humanitarian purposes in Ukraine that result from Russian aggression. Further details will be set out in due course."
Meanwhile, Chelsea expressed their delight that a process which usually takes nine months could be completed in three.
A Blues statement read: "The Club received more than 250 enquiries from proposed purchasers, held detailed discussions with more than 100 individuals and entities, and entered into 32 confidentiality agreements, which allowed for due diligence with respect to confidential Club information.
"Ultimately, the Club received 12 credible bids, resulting in 4 and then 3 final bidders. The Todd Boehly and Clearlake Capital consortium was chosen as the preferred bidder."
Confirmation of the takeover has been in the pipeline for some weeks and Abramovich had already bid an emotional farewell to the club on Saturday.
He said: "It has been nearly three months since I announced my intention to sell Chelsea FC.
"During this time, the team have worked hard to find the right custodian for Chelsea FC that would be best positioned to successfully lead the Club into its next chapter.
"I am pleased this search has now come to a successful conclusion. As I hand over Chelsea to its new custodians, I would like to wish them the best of success, both on and off the pitch.
"It has been an honour of a lifetime to be a part of this Club - I would like to thank all the Club's past and current players, staff, and of course fans for these incredible years."