Over 1,300 investors cast their vote on Tuesday with 80.28% in favour.
The offer needed 75% approval to go ahead.
Chairman Russell Scrimshaw said: “The positive outcome from today’s meeting secures a return for shareholders, and provides greater certainty in terms of safeguarding the project, protecting the jobs of our employees, and allowing the community, region and the UK to continue to benefit from the project.”
The offer remains subject to approval by the competition authorities.
Investors met on Tuesday in London in an angry mood to confront the board, which has recommended the takeover since Anglo American approached them.
Directors replied it would have been the only viable solution for the project to carry on, pointing out Anglo could still buy Sirius from administrators if, as they suggested would happen, it went bankrupt should the deal not be approved.
The polyhalite mine needs another US$1.3bn to be built, the directors said, although a strategic review carried out last September showed the firm needed £460mln to carry on for two years.
A group of investors had been protesting on those grounds, saying it could have provided that cash through bonds to avoid selling the company cheaply.
Many investors who will receive 5.5p per share from Anglo had bought their holding at four times that price, as Sirius was trading at 20p just a year ago. In lunchtime trade on Wednesday, Sirius shares were changing hands at 5.49p each, up 17% on Tuesday's closing price.
The North Yorkshire firm was hit by funding problems after the government declined to bail the project out
On Tuesday, the board said that Westminster simply chose not to help.
“As much as the Prime Minister makes these statements about wanting to support the north, this is the poster child for what they should be supporting and they chose not to,” chief executive Chris Fraser said.
Tuesday’s meeting was a battleground between the board and some investors who were not ready to give up the “Sirius dream”, with some admitting they had been carried away by emotions.
The directors apologised for the eventual outcome but reiterated that the deal was the only way forward.
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