Spurs's first victory at Old Trafford since December 1989 saw them twice go two goals in front, with the teams sharing three goals in as many minutes during a frantic spell before the hour mark.
United's first home league defeat in 2012 meant Sir Alex Ferguson's side ended the weekend in third place - four points behind leaders Chelsea - while Tottenham climbed to fifth.
Just 90 seconds had been played when Spurs' defender Jan Vertonghen burst into the box and fired the visitors in front with a low shot that deflected off Jonny Evans.
In the 32nd minute they were 2-0 up when Bale, from just inside the United half, outpaced Rio Ferdinand before his shot beat Anders Lindegaard. United pulled a goal back in the 51st minute when substitute Wayne Rooney provided a cross for Nani to steer past Brad Friedel.
But two minutes later Jermain Defoe left the struggling Ferdinand behind him and found Bale. The Welshman's shot was palmed away by Lindegaard but only to Clint Dempsey, with the USA striker shooting into the empty net.
However, United cut the lead to 3-2 moments later when, after a superb pass from Robin van Persie, Japan midfielder Shinji Kagawa slotted home.
Rooney hit the post and Michael Carrick's header came off the crossbar as United pressed for an equaliser, but Spurs held on for a famous victory.