The card, from the Majestic Hotel in Belgrade, was signed by 14 members of the squad on 5 February 1958.
The sale in Ludlow, Shropshire, had been expected to fetch about £1,500.
John Mullock of auctioneers Mullock Madeley said: "We expected an awful lot of interest as there had been nothing like this on the market before."
He added: "It's so rare, it's one of the most important items of football history relating to one of the worst disasters ever to hit the sporting world."
The disaster happened as the United team was travelling back from a European Cup tie in Belgrade.
Among the 23 who died in the crash were eight members of the famous Busby Babes - named after manager Sir Matt Busby - Geoff Bent, Roger Byrne, Eddie Colman, Duncan Edwards, Mark Jones, David Pegg, Billy Whelan and Tommy Taylor.
All eight signatures are included on the rear of the menu card, which was acquired by the seller's father, who was the British Ambassador to Belgrade.
It was the ambassador who introduced the players on the pitch before the game against Red Star Belgrade in February 1958.
The air disaster claimed the lives of 23 people connected to the club, including players, officials and journalists.
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