Togo striker Adebayor opened the scoring for Tottenham in the 34th minute when he met an Eriksen cross, becoming only the third player to score in three consecutive games at Old Trafford.
Eriksen doubled their lead in the 66th minute with a well-placed header after beating his marker, but Danny Welbeck brought United back into the game with his seventh goal of the campaign a minute later.
That set up a tense finish, and United pushed hard for an equaliser and threw everything forward in the closing stages.
But chances for substitute Javier Hernandez, Adnan Januzaj and Welbeck were all wasted as United slipped to seventh - 11 points behind leaders Arsenal - and saw their Premier League title hopes suffer a potentially fatal blow.
Boss David Moyes made four changes in attacking positions to the United line-up from their victory over Norwich, with Wayne Rooney and Welbeck returning up front.
Kyle Walker and Danny Rose returned in the full-back positions for the visitors, with Etienne Capoue replacing the injured Paulinho in midfield.
Welbeck and Chris Smalling both had early chances for the hosts as United attacked down the right, but Hugo Lloris made good saves to keep Tottenham in the game.
The visitors had a fantastic chance to open the scoring in the 14th minute through Aaron Lennon, but the winger wasted a perfect throughball from Roberto Soldado as he fired straight at David De Gea.
With their next attack, in the 34th minute, Sherwood's side opened the scoring through Adebayor with his third league goal of the season.
Eriksen raced down the right before delivering a perfect cross towards the back post, where Adebayor rose unmarked to score.
They should have doubled their lead five minutes later with another attack on the right.
This time, Lennon raced past Patrice Evra and played a low cross into the unmarked Soldado, but the Spaniard kneed the ball wide from six yards.
After a quiet start to the second half, the visitors doubled their lead in the 66th minute as Eriksen netted his second goal in three games.
The Dane anticipated the flight of a deflected Lennon cross far better than Antonio Valencia, stealing in front of the United man to head past De Gea from five yards.
However, United halved the deficit immediately through Welbeck.
Januzaj played a neat pass behind the Tottenham defence for the striker, and he latched onto the ball and calmly lifted it over Lloris.
The goal woke United up and Januzaj had a fantastic chance to level the game late on, but his long-range effort curled just wide of Lloris' right hand post.
That sparked a flurry of late chances to level, the best of which fell to Rooney and Nemanja Vidic, but they both found Lloris in excellent form as he saved well.
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