But the announcement, which caused market chaos and damaged investors, had no basis in fact, the SEC said yesterday.
The SEC had initially sought to ban Mr Musk, 47, from working on the board of any publicly traded company.
However, under a proposed deal, Mr Musk will remain Tesla CEO but steps down as chairman for three years.
Both he and Tesla also have to pay a fine totalling $40million (£30.6m).
Tesla must also appoint an independent chairman, two independent directors, and a board committee to control Mr Musk’s communications.
Mr Musk’s tweet on August 7 is one of a number of controversial incidents this summer which have brought into question the judgment of the founder and CEO of space exploration company SpaceX.
He is being sued for $75,000 (£57,000) in a defamation action after allegations of child abuse against British diver Vernon Unsworth, who was involved in the rescue of 12 Thai teenagers from a flooded cave in July.
Early last month Mr Musk appeared on a podcast smoking marijuana, causing Tesla shares to fall by more than nine per cent, even though the drug is legal in California, where the recording was made.
In a cryptic tweet two days ago, Mr Musk said: “That was intense.”
SEC chairman Jay Clayton said: “The prompt resolution of this matter on the agreed terms is in the best interests of our markets and our investors, including the shareholders of Tesla.”
The AA’s president Edmund King described Mr Musk as a “visionary” – and said Tesla might even thrive as time went on.
He said: “Losing someone who’s been their talisman will cause hiccups, but if they can get leadership which is not quite as maverick, Tesla could be in a very good long-term position.
“There’s still a great demand for the Tesla Model 3, which is oversubscribed, and the market is only going to increase, especially when the UK Government has said it wants to get rid of petrol and diesel cars by 2040. That’s a move governments across the world are making.”
Business expert Ivan Feinseth of Tigress Financial Partners called the SEC penalty a “slap on the wrist” for Mr Musk, adding: “The fact that he can remain CEO is very important for the company.”
It was not clear who would replace Mr Musk as Tesla chairman.