The stock market's Takeover Panel, which governs merger and acquisition deals in the UK and arranged the auction, said Fortress offered 286p per Morrisons ordinary share, while CD&R offered 287p - meaning its bid amounts to an offer of almost £7bn.
The final offer for the supermarket will now be voted on by shareholders on October 19.
Andrew Higginson, chairman of Morrisons, said the final offer from CD&R "represents excellent value for shareholders" and protects the "fundamental character of Morrisons for all stakeholders".
He added: "CD&R have good retail experience, a strong record of developing and growing the businesses in which they invest, and they share our vision and ambition for Morrisons.
"We remain confident that CD&R will be a responsible, thoughtful and careful owner of an important British grocery business.
"Shareholders will now have the final say and, if the offer is approved, the board is confident that Morrisons will continue to go from strength to strength under CD&R's ownership."
Sir Terry Leahy, the former chief executive of Britain’s biggest supermarket chain Tesco, is a senior adviser to CD&R.
He will be reunited with Morrisons CEO David Potts and Mr Higginson, two of his closest lieutenants at Tesco.
Sir Terry said: "We are gratified by the recommendation of the Morrisons' board and look forward to the shareholder vote to approve the transaction.
"We continue to believe that Morrisons is an excellent business, with a strong management team, a clear strategy, and good prospects."
The takeover saga has dragged on since CD&R first made an approach for the Bradford-based grocer back in June, leading to speculation the sector was ripe for private equity takeovers.
Following the initial bid, rival Softbank-backed Fortress made an offer of £6.3bn in July.
But shareholders felt this was too low and Fortress, which owns Majestic Wines, returned with an increased offer of £6.7bn in August, which the board accepted.
Later that month CD&R, which boasts former Tesco boss Sir Terry Leahy as an adviser, returned with an increased bid of £7bn.
This led to the board withdrawing its support for the Fortress bid and throwing its weight behind the higher offer.
But because neither side made a formal bid, the Takeover Panel launched an auction process.
Joshua A. Pack, managing partner of Fortress, said: "Morrisons is an outstanding business and we wish the company and all those involved with it the very best for the future.
"The UK remains a very attractive investment environment from many perspectives, and we will continue to explore opportunities to help strong management teams grow their businesses and create long-term value."
Both sides agreed beforehand that all bids would be at a fixed cash price and could not include stakes in other businesses or dividends to shareholders.
Both sides have been keen to stress they want to uphold the supermarket's values and have attempted to ward off suggestions they will start selling off vast swathes of the company's freeholds.
Supermarkets typically lease properties, whilst Morrisons continues to own around 90% of its estate.
There have also been concerns that any new owner may reduce the supermarket's tax bill, with off-shore shell companies set up ahead of the takeover.
Morrisons' pension trustees will have to be consulted, although earlier this month they said an agreement had been reached with CD&R.
Seema Malhotra, shadow minister for business and consumers, said: "The new owners must urgently deliver binding assurances for workers, pension fund holders, and local people.
"Government has a responsibility too. It must ensure that the new owners are responsible, long-term investors, seeking to build the business for the future, and that decisions taken are also in the public interest.
"We cannot see a repetition of previous cases where businesses have been loaded with debt and asset-stripped. Morrisons is a great British company which must be safeguarded for the future."
Joanne McGuinness, Usdaw national officer, said: "Now that the future ownership of Morrisons seems clearer we are seeking discussions with the prospective new owners, to ensure that our members' interests and the long-term future of the business are protected throughout this process.
"Our main concerns are about job security, while maintaining and improving our members' terms and conditions of employment.
"We have heard the assurances already given and welcome the constructive working relationship that Usdaw has experienced so far."
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