Arsenal's worst start to a season since 1994 looked set to continue after Gilberto Silva's early penalty was saved by debutant goalkeeper Tomasz Kuszczak, but Adebayor struck after 85 minutes to give the Londoners their first victory.
The Gunners had failed to score in five of their previous six games against their old rivals, but the impressive Togo striker eased that drought when he ran on to a Cesc Fabregas pass and finally managed to beat Kuszczak.
Poland international Kuszczak, deputising for the unwell Edwin van der Sar, had previously repelled everything Arsenal threw at him, starting with his 12th-minute save from Gilberto Silva after he had fouled Adebayor to concede the penalty.
Arsenal were again without injured captain Thierry Henry and played with just Adebayor up front, but that did not stop them punching regular holes in the home side's defence.
Moments after the penalty miss Adebayor's header was pushed on to the post by the athletic Kuszczak and the 24-year-old then made a flying save to keep out a powerful shot from Tomas Rosicky.
United struggled to get a foothold in the game in the opening half and were almost solely reliant on Cristiano Ronaldo for their attacking threat.
The Portugal international had twice gone close before he thought he had opened the scoring after 43 minutes, only for his close-range volley to bounce to safety off the face of Arsenal goalkeeper Jens Lehmann.
United improved slightly after the break and leading scorer Louis Saha headed a Darren Fletcher cross just wide.
But the visitors continued to look the better side and posed a constant danger on the break.
Kuszczak again had to deny Adebayor and United were given a further let-off when Julio Baptista missed a chance after John O'Shea had conceded possession.
Alex Ferguson's side did not learn from their mistakes, though, and when Fabregas took the ball off Ronaldo he threaded through a pass for the highly impressive Adebayor to beat Kuszczak at full stretch.
It took a superb late save from Lehmann to stop Ole Gunnar Solskjaer from grabbing a dramatic equaliser but Arsenal fully merited a victory which suggests they are not a spent force on the domestic front on what proved to be another good day for champions Chelsea following their earlier win over Liverpool.
