He tweeted “the bird is freed,” in an apparent reference to the deal closing.
A number of top executives, including the boss, Parag Agrawal, have reportedly been fired.
It brings to a close a saga that saw Twitter go to court to hold the multi-billionaire to the terms of a takeover deal that he had tried to escape.
Twitter has not yet confirmed the takeover, but an early investor in the company told the BBC that the deal had been completed.
Mr Musk, a self-styled “free speech absolutist”, has been critical of Twitter’s moderation policies and the news will be greeted with mixed feelings by Twitter users and employees.
Many people on the right of US politics will celebrate the exit of Mr Agrawal as chief executive. They view people like Mr Agrawal, and his predecessor, Jack Dorsey, as liberals who are curtailing free speech.
They also think that under their stewardship, Twitter has censored conservative voices – an accusation that Twitter denied.
Mr Agrawal, chief financial officer Ned Segal, and the firm’s top legal and policy executive, Vijaya Gadde, are no longer with the company, according to US media reports.
Mr Agrawal and Mr Segal were escorted out of Twitter’s San Francisco headquarters after the deal closed, the Reuters news agency reported
Twitter co-founder Biz Stone thanked Mr Agrawal, Mr Segal and Ms Gadde for their “collective contribution” to the business.
Meanwhile, Bret Taylor – who had served as Twitter’s chairman since last November – updated his LinkedIn profile to indicate that he was no longer in the post.
It is not clear yet if the reported board clear-out is the opening salvo in company-wide job cuts.
Ross Gerber, president and chief executive of Gerber Kawasaki Wealth and Investment Management, who is a shareholder in both Twitter and Mr Musk’s other company Tesla, told the BBC reports that 75% of staff at the social media company will lose their jobs is “inaccurate”.
“The figure that they have mentioned originally was around 50% of the staff,” he said. “But that was predicated on the teams that were working on reviewing who the staff was and how good the staff is.”
He said it is likely Twitter’s new owner will want to keep people “especially on the engineering side”.
“Really what they’re looking at from the trimming side is management which they’ve already started with upper management,” said Mr Gerber. He said cuts are then likely to extend to product managers “and products they’ve been working on that aren’t going anywhere”.
The social media platform’s shares will be suspended from trading on Friday, according to the New York Stock Exchange’s website.
Mr Musk said he bought the social media platform to help humanity and he wanted “civilisation to have a common digital town square”.
Earlier this week, Mr Musk tweeted a video of himself walking into Twitter’s headquarters in San Francisco carrying a kitchen sink with the caption: “let that sink in!”
He also changed his Twitter profile to read “Chief Twit”.
