The game was only 10 minutes old when Villarreal midfielder Marcos Senna received the ball outside the area and scored with a fierce drive that flew into the top corner.
Arsenal were hindered by first-half injuries to goalkeeper Manuel Almunia and defender William Gallas who both had to be substituted, as Villarreal dominated.
But the Londoners responded in the second half and Adebayor’s acrobatic effort from Cesc Fabregas’s pass in the 66th minute left the visitors with an away-goal advantage for the return leg in London on April 15.
"I don’t even know how I scored that, but the most important thing was that the ball went into the back of the net," Adebayor told ITV.
"It was quite difficult for us because they played some very good football. The second leg is going to be tough, but we will do everything to get through. We got a good result to take home and we are very happy."
A match which started at a hectic pace, full of slick passing from both sides, was blown open when Senna’s thunderbolt left Almunia with no chance.
Visiting defender Gael Clichy had already blocked an attempt from a corner on the line by then, but Arsenal responded with Samir Nasri testing keeper Diego Lopez at his near post in the 18th minute.
Arsenal manager Arsene Wenger was forced into making a change in the 27th when Almunia had to be substituted for Lukasz Fabianski, after he failed to recover from an earlier knock to his right ankle.
Fabianski had barely got his bearings when he had to make a a double save from Senna and then Joan Capdevila.
Three minutes before the break, it was Gallas, struggling with an injury to his right leg, who had to be replaced by Johan Djourou as Villarreal comfortably held the visitors at bay.
Arsenal playmaker Fabregas and winger Theo Walcott started to find some space after the break, though it was not until midway through the second half that Adebayor threatened the home goal with a header.
Soon after, Fabregas’s long pass found Adebayor with his back to goal in the area, and the Togo striker controlled on his chest and twisted in the air to volley over his shoulder, and wide of Lopez into the far corner.
Villarreal, who are unbeaten at home in this competition, pressed hard towards the end with Senna and Diego Godin going close, but Arsenal held firm and carried plenty of threat themselves on the break.
Senna told Antena 3 television: "We started at a good pace but eased off a little in the second half. That’s the way quarter-finals go sometimes.
"The tie remains open. Although they have home advantage now and an away goal we won’t be throwing in the towel."