At the end of a shareholder meeting Thursday, Dell officials said that based on preliminary results, the proposal had enough votes in favor of it to pass. The company did not immediately announce the tally.
"I am pleased with this outcome and am energized to continue building Dell into the industry's leading provider of scalable, end-to-end technology solutions," Michael Dell, the company's chairman, CEO and founder, said in a statement.
Dell was present for the meeting, which lasted about 15 minutes. About 100 people were in attendance, though few appeared to be rank-and-file shareholders. The meeting ended with a light applause after the approval was announced.
Like other PC makers, Dell Inc. has been hit hard in recent years as consumers shift their buying habits away from traditional desktops and laptops and toward tablets and other mobile devices.
Last month, Dell reported a 72 percent drop in profit for its most recent quarter, as the company cut prices to shore up computer sales. Dell's stock has plunged by more than 40 percent since Michael Dell returned for a second stint as CEO in 2007. On Thursday, Dell's stock gained a penny to $13.86.
Michael Dell, who made his offer with an investment group led by Silver Lake Partners, has said he can turn the company around. But he has said the process will involve a painful realignment that is likely to trim its earnings for another year or two. As a result, he believes, the turnaround will be easier to pull off away from Wall Street and its fixation with short-term results.
The deal is expected to be completed within two months. The company will continue to be based in Round Rock, Texas.
Critics of the offer said it undervalued the company. The vote was delayed three times as a result of the opposition.
Before the last delay, Michael Dell and Silver Lake agreed to pay a special dividend of 13 cents per share to supplement a bid that had already been raised from $13.65 per share to $13.75 per share.
Despite the enhanced offer, activist investor Carl Icahn and investment fund Southeastern Asset Management continued to contend that the company was worth more than what was being offered.
Icahn dropped his opposition on Monday, saying that while he still opposed the sale, it would be "almost impossible" to defeat the offer in Thursday's vote.
Bree Fowler reported from New York.