Christian Eriksen doubled Spurs' lead, after Adebayor had claimed his fourth goal in five games under new manager Tim Sherwood, before Danny Welbeck replied in vain for United.
Adebayor was stretchered off in the 70th minute, but not even that injury could take the shine off a Spurs performance that left United in seventh place and trailing leaders Arsenal by 11 points.
Tottenham were required to weather a bright and breezy opening from their hosts, but by the time Adebayor struck his 34th-minute opener, the north London club were good value for their lead.
The goal came following a blistering counter-attack as Kyle Walker broke up a United foray and Spurs exploited space on United's left.
Eriksen picked out Adebayor with a superb cross that the Togo striker met, outjumping Chris Smalling in the process, to place an accurate header beyond the dive of David de Gea.
Adebayor, revelling under the new White Hart Lane management, almost had a hand in a second goal minutes later as he won the ball in midfield and found Aaron Lennon, whose cross was put wide by a sliding Roberto Soldado.
A two-goal advantage would not have flattered Tottenham, with Lennon squandering arguably the best chance of all as he raced onto Soldado's 14th-minute pass, only to be denied by De Gea.
It was a stark contrast to the opening exchanges, when Tottenham goalkeeper Hugo Lloris appeared fortunate to escape after seeming to handle the ball outside his box as he thwarted a United attack.
Welbeck was then denied by the Frenchman and Lloris enjoyed a slice of luck when he saved Smalling's fierce shot with the inside of his thigh before the rebound squirmed over the byline for a corner.
But for all their early possession and swift attacking play, United failed to really test Lloris in the opening hour.
Wayne Rooney's frustration earned him a 41st-minute booking for a tug at Mousa Dembele, while Welbeck unsuccessfully attempted to earn a penalty after Vlad Chiriches made slight contact with him.
Appreciating the dire situation, United manager David Moyes made a bold double substitution on the hour, bringing on forwards Shinji Kagawa and Javier Hernandez for midfielder Michael Carrick and full-back Smalling.
The move saw Antonio Valencia switch to right-back and it back-fired when he responded slowly as Spurs made it 2-0 in the 66th minute.
Again, it was a goal carved out of a counter-attack, with Soldado slipping the ball inside for Lennon, whose chipped pass across the area was met by Eriksen's stooping header as Valencia was found wanting.
Fortunately for Moyes, however, it took less than a minute for Welbeck to respond as he chased Adnan Januzaj's perceptive pass between two defenders and finished confidently over the advancing Lloris.
Januzaj led the charge for the equaliser, shooting just wide from 20 yards then slaloming in from the right before crossing to the near post, where Hernandez's back-heel was blocked by the goalkeeper.
Lloris also showed a rush of blood in chasing out of his area to challenge Ashley Young, sparking United penalty appeals, although the goalkeeper made amends with superb stops in quick succession from Rooney and Nemanja Vidic.