Rafik Djebbour scored after 11 minutes when he slotted home Ignacio Scocco's inviting cross in the first leg of the qualifying round tie.
After a nervous opening the hosts found their composure as the game wore on but were unable to properly test the AEK rearguard, although Jon Daly could have scored a precious equaliser in the closing stages but failed to find the target with only the goalkeeper to beat.
Dundee United made one change with youngster Keith Watson preferred to defender Darren Dods, while there was no place in the squad for influential midfielder Danny Swanson, who had been struggling to recover from an Achilles injury.
AEK were without defender Daniel Majstorovic following his move to Celtic but the visitors still fielded a strong side which included Brazilian Leonardo and highly-rated Argentinian winger Scocco.
However, former Fulham and Portsmouth midfielder Papa Bouba Diop had to be content with a place on the bench.
AEK settled into the game immediately and were content to hold on to the ball and play a slow, passing game, while the hosts made a nervy start and were fortunate that a couple of stray passes went unpunished.
But the hosts were made to pay after 11 minutes when Scocco's tantalising ball across the United goal was slammed into the back of the net by the clinical Djebbour.
The same pair combined three minutes later but this time Dusan Pernis was equal to Djebbour's low drive.
United created their first opening soon after but Daly headed over from a David Goodwillie cross.
Peter Houston had warned his players about strict referee Marijo Strahonja but Craig Conway failed to take heed of his manager's words and was booked after 18 minutes for throwing the ball away.
As the half wore on United came more into the match but were too often let down by a poor final ball.
Paul Dixon did, however, send an inviting cross into the area after 32 minutes but goalkeeper Sebastian Saja punched the ball to safety.
Nikos Liberopoulos was booked just before the break for a crude challenge on Conway, who was United's most promising outlet during the opening period.
AEK made a lively start to the second half with Leonardo unleashing a low drive from the edge of the area that Pernis gathered at the second attempt.
Conway created an opening for himself with some fancy footwork but his powerful shot was blocked by Pantelis Kafes.
Leonardo should have done better after 56 minutes but the Brazilian could only divert his header straight at Pernis.
United continued to press for an equaliser and the ball broke kindly on the edge of the area for Watson after 62 minutes but the young right-back failed to keep his volley under the target.
The visitors introduced the speedy Nathan Burns for the injured Leonardo as they looked to conjure up the second goal that would effectively kill off the tie as a contest and with 10 minutes left to play the Australian was inches away from turning home a Scocco cross.
Irishman Daly, who had been starved of service for most of the evening, could have restored parity in the closing stages when the ball broke for him in the area but he dragged his left-foot shot woefully wide of the target.