Musk, who made the shock bid less than two weeks ago, has claimed he is the right person to "unlock" the social media firm's "extraordinary potential".
He has suggested a series of changes from relaxing its content restrictions to eradicating fake accounts.
The firm initially rebuffed Musk's bid, but it will now ask shareholders to vote to approve the deal.
Musk is the world's richest man, according to Forbes magazine, with an estimated net worth of US$273.6b mostly due to his shareholding in electric vehicle maker Tesla which he runs. He also leads the aerospace firm SpaceX.
"Free speech is the bedrock of a functioning democracy, and Twitter is the digital town square where matters vital to the future of humanity are debated," he said in a statement announcing the deal.
"I also want to make Twitter better than ever by enhancing the product with new features, making the algorithms open source to increase trust, defeating the spam bots, and authenticating all humans.
"Twitter has tremendous potential - I look forward to working with the company and the community of users to unlock it."
Bret Taylor, chair of Twitter's board, said the board had conducted a "comprehensive process" to assess Musk's proposals for the firm.
"The proposed transaction will deliver a substantial cash premium, and we believe it is the best path forward for Twitter's stockholders," he said.
Twitter's shares were up about 6 percent following the news.