The Dane capped a rapid counterattack with a brave header to double the visiting side's lead and although Danny Welbeck pulled one back a minute later, Tim Sherwood's men held on.
Togo striker Adebayor opened the scoring in the 34th minute when he met an Eriksen cross, becoming only the third player to score in three consecutive games at the home of the Red Devils.
Eriksen doubled the 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 just a minute later.
That set up a tense finish as United pushed hard for an equalizer and threw everything forward in the closing stages.
But chances for substitute Chicharito, Adnan Januzaj and Welbeck were all wasted as United slipped to seventh - 11 points behind leaders Arsenal - and saw its Premier League title hopes suffer a potentially fatal blow.
David Moyes made four changes in attacking positions to the United lineup from its victory against Norwich, with Wayne Rooney and Welbeck returning up front.
Kyle Walker and Danny Rose returned in the fullback 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 through ball from Roberto Soldado as he fired straight at David De Gea.
With its next attack, in the 34th minute, Sherwood's side opened the scoring through Adebayor netting his third league goal of the season.
Eriksen raced down the right before delivering a perfect cross toward the back post, where Adebayor rose unmarked to score.
It should have doubled its 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 defense 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, 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.