Spanish midfielder Cesc Fabregas set up Adebayor for his goal in a match that Arsenal deserved to win after outplaying their arch rivals who had won four from four to start the season.
The 1-0 win was a huge boost for the Gunners, prompting manager Arsene Wenger to remind critics that his team are still very much premiership contenders.
"Nearly everyone has written us off after the start we have made. If you open the newspapers we will be playing in the (Coca-Cola) Championship next season," Wenger said.
"I hope that this result sends out the right message. We do want to win the title.
"You can see we are a strong side from today's performance. I believe in this side. We controlled the game and we were always more dangerous than them," he added.
Without star striker Thierry Henry through injury, Adebayor played a lone role up front for Arsenal, but was well supported by Alexander Hleb, Tomas Rosicky and Freddie Ljungberg who all made incisive input from midfield, with the tireless Fabregas running all match.
For United, midfielder Cristiano Ronaldo took a while to get going, and even then was perhaps short of his best, while the returning Wayne Rooney and Paul Scholes failed to make any impact.
Early penalty saved
In the 10th minute, Manchester United goal keeper Tomasz Kuszczak who was deputising for the ill Edwin van der Sar fouled Adebayor giving away a penalty, but atoned by saving the resulting spot kick from Gilberto Silva.
"To miss a penalty in a match of this size and still come back and win shows the character we have in the team," said Wenger.
"This team has big potential and we showed great spirit."
Sir Alex Ferguson, Red Devils manager, conceded that Arsenal were the better team on the day.
"We looked tired in the second half and Arsenal deserved their victory," Ferguson said.
"Arsenal were much stronger in the last 25 minutes. It was a bad performance and we can do better."
Other Sunday matches
In other Sunday action, Newcastle stunned West Ham with a 2-0 victory at Upton Park through second half goals from new signings Damien Duff and Obafemi Martins.
Tottenham and Fulham played out a 0-0 draw at White Hart Lane while in contrast, six goals were scored at Ewood Park in Blackburn's 4-2 victory over Manchester City.
City midfielder Trevor Sinclair scored an own-goal to opening first half scoring, with Joey Barton hitting an equalizer in the 38th minute. Blackburn went ahead once more through Morten Gamst-Pedersen only to see defender Andre Ooijer score an own-goal of his own to make the scores 2-2 at half time.
Rovers ran away with the match in the second half with goals from Benni McCarthy and Paul Gallagher in the 65th and 88th minutes respectively.
Earlier, Chelsea defeated Liverpool 1-0 with Didier Drogba hitting a wonder strike in the first half, and Michael Ballack being sent off with a straight red card after stamping on the leg of midfielder Mohamed Sissoko in the 51st minute.