20-year-old Shaw will have an operation on his return to Manchester, the club said.
He was treated on the pitch for 10 minutes after Hector Moreno's tackle.
"It was a very bad tackle. He had the oxygen mask on and was crying in the dressing room," said boss Louis van Gaal.
Shaw has started all of United's eight games this season, as well as playing in England's European Championship qualifiers against San Marino and Switzerland.
Yet with just nine months to go until Euro 2016, Shaw has faces a tough battle to get himself fit and ready in time to get into the England squad again.
Less than an hour after suffering the injury, Shaw wrote on Twitter: "Thank you everyone for your messages, words can't describe how gutted I am, my road to recovery starts now, I will come back stronger."
Memphis Depay scored the opener for Manchester United before Hector Moreno, who has apologised for his tackle on Luke Shaw, headed in an equaliser just before half-time.
Luciano Narsingh then nodded home in the second half to secure victory for PSV Eindhoven.

All 25,000 seats for the four-night event in the grounds of Windsor Castle had been bought within five hours of going on sale at 09:00 GMT.
It will take place between May 12 and May 15 next year, with members of the Royal Family due to attend each night.
The Queen herself will attend a performance on the final evening, which will be broadcast live on ITV.
Singer Katherine Jenkins will be among 1,500 participants including musicians, dancers, famous actors and artists. The event will also feature 900 horses.
Tickets were priced from Â£55 to Â£195, with the most expensive seats situated next to the Royal Enclosure.
Some 5,000 free tickets will be available which will allow guests to see celebrities arriving on the red carpet. Details of how to apply will be announced in the new year.
The final night's show was the first to sell out. The Queen turns 90 on 21 April 2016.
A spokeswoman for the event organisers said: "They were selling very, very quickly. By 9.30am, 4,000 tickets had already been sold."
The celebration at Windsor is a not-for-profit event and aims to raise a surplus to donate to charities.
National commemorations will centre on her official birthday weekend in June with a mass street party in The Mall, a service of thanksgiving at St Paul's Cathedral and the traditional Trooping the Colour ceremony.
Defender Luke Shaw broke his right leg in two places in Manchester United's 2-1 Champions League loss at PSV Eindhoven in the Netherlands.
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Tickets for a celebration to mark the Queen's 90th birthday sold out within hours of going on sale.