Michael Dell has the won the battle for control of the computer company that he created, after shareholders backed his $24.8bn (£15.7bn) offer to take Dell private and revive the struggling business away from the incessant pressure of Wall Street.

The vote clears the way for the huge buyout, in which Dell is working with private equity partners Silver Lake after seeing off a challenge from activist investor Carl Icahn.

The move comes as traditional PC companies struggle to make money in a market that is rapidly shrinking, with the latest forecasts showing that PC sales will be overtaken by tablets for the first time in the final quarter of this year.

Research group IDC said tablets will outsell all PCs in the three months to Christmas, and by 2015 they will regularly outsell PCs as consumers' tastes shift towards the more mobile format.

"We are forecasting 84.1m tablets will be shipped in the fourth quarter of 2013, compared to 83.1m PCs," said an IDC spokesman.

The abruptness of the shift – modern tablets only went on sale with the launch of Apple's iPad in April 2010 – has caught a number of traditional PC companies, including Dell, flatfooted and with no viable strategy in the fast-growing mobile market.

By 2017, says IDC, 87% of devices connected to the internet will be tablets and smartphones, relegating PCs from their former position as the main means of getting online.

Speaking after the Dell vote, Francisco Jeronimo, IDC's smartphones analyst, said the wider market shift will "kill many old companies and create new leaders".

PC makers such as Dell have struggled with limited profits for years. But as consumers, who made up half of PC buyers, have shifted spending to tablets and smartphones, the industry has suffered falls in sales which are squeezing small and large players alike.

Michael Dell, who founded the company from a college dorm room in 1984, has fought for months to convince investors his offer is the best option for the company's future. Icahn ended his challenge after deciding it was "impossible to win".

Dell has been forced into a buyout after 25 years on the stock market because the PC market is dwindling so rapidly. The company reported a 72% slump in quarterly earnings last month as it cut prices and prepared an attack on the market for supplying big "enterprise" businesses.

The iPad was rapidly followed by more tablets, from companies including Samsung, Amazon and Google. After years when PC sales growth was regularly between 10% and 20%, it has gone into reverse for the past five quarters, and IDC forecasts a drop of nearly 10% this year.

Other companies besides Dell have been hit due to the lack of a successful tablet strategy. Taiwan's Acer, one of the five biggest PC makers, has seen profits collapse since 2011, and revenues have shrunk for five quarters in a row. HP, which bought Palm for $1bn but wrote off its TouchPad tablet in August 2011 after less than two months, is also struggling to make a profit from PCs.

Michael Dell argues that the company needs to go private so it can restructure to cope with the new computing landscape away from the quarterly pressures of Wall Street, and focus as IBM does on providing its storage, servers and services to public organisations and large companies.

Dell first proposed the buyout in private discussions in June 2012. It was announced on 5 February at $13.65 per share. Rival bidder Blackstone raised that to $14.25 in March, and Icahn offered $15 soon afterwards. Dell finally offered $13.75 per share.

A vote on the buyout had been postponed three times as Michael Dell and the company's board scrambled to garner enough votes in favour. But last month Dell raised his offer price, tacked on a special-dividend sweetener, and got the board to change voting rules so that abstentions no longer count against him – turning the tide in his favour.

A change of over two months in the deadline for proxy votes may also have brought backing from hedge funds that bought shares for short-term gain and were likely to support a buyout.