The club was put up for sale in March before previous owner Roman Abramovich was sanctioned over his links to Russian President Vladimir Putin.
Chelsea had been operating under a special government licence which would have expired on 31 May.
Boehly said in a statement that he was “honoured” and “wanted to make fans proud”.
His consortium fought off 11 serious rivals to become the new owners, in a sale process that started on 2 March and comprised more than 250 enquiries.
“We are honoured to become the new custodians of Chelsea Football Club,” he said.
“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.”
The UK government – which said last Wednesday it could issue a licence for the sale of the club – does not want Abramovich to receive any of the proceeds from the sale, which will instead go into a frozen bank account to be donated to charities supporting victims of the war in Ukraine.