After a week of farce which saw the venue for the match switched several times, the contest finally went ahead at the Karaiskaki Stadium - home of AEK's deadly rivals Olympiakos - but with only United supporters present because of security fears.
Any hopes United, who were trailing 1-0 from the first leg, had that the eerie atmosphere would unsettle AEK were quickly dashed as they picked up where they left off at Tannadice with an impressive passing game.
Debutant Papa Bouba Diop scored the opening goal and it looked as if United's hopes were over, but substitute Jon Daly came off the bench to net a goal which gave them hope of progressing. Despite their endeavour, though, it was not to be.
Houston pulled a major shock pre-match by dropping strikers Daly - his captain - and David Goodwillie. In came David Robertson and also Danny Cadamarteri, who was recently told he is available for free transfer.
AEK were without the services of their two stand-out players from the first leg, Nikos Liberopoulos and Traianos Dellas, who picked up injuries in training so were unable to play.
Former Fulham midfielder Diop was drafted in for his first game.
With United's 500 fans safely inside the ground after being bussed to the stadium amid tight security, they were intent on being vociferous.
Their good mood was almost shattered inside four minutes, though, when Leonardo broke through into the penalty area but goalkeeper Dusan Pernis saved with his feet.
Panagiotis Lagos was next to have a go for the Greeks but his effort from 25 yards sailed over the crossbar.
United were fortunate to stay on level terms after 10 minutes when Ignacio Scocco picked up a Rafik Djebbour ball inside the penalty area but somehow fired over from just six yards.
The Tangerines broke upfield on 20 minutes and looked like they were set to cause AEK problems when Scott Robertson fed Cadamarteri, but he strayed offside.
That moment proved pivotal as AEK swiftly took the lead through Diop. United defender Sean Dillon conceded a corner kick needlessly and they were punished as Sanel Jahic rose unmarked to flick the ball on for the Senegal man, who poked a shot in from four yards.
United came close to equalising on the half hour when Keith Watson picked out Craig Conway at the back post with a low cross from the right, but the linesman chalked his finish off for offside.
Houston was forced into a change just minutes into the second half when Cadamarteri pulled up injured and was replaced by Goodwillie.
In an attempt to inject new life into his side the United boss brought on Danny Swanson, back after a spell out injured, for David Robertson.
Houston then introduced Daly in place of Gomis in a bid to find the goal they needed to get back into the contest.
AEK could have scored on 73 minutes when Scocco got in behind the defence but again Pernis was able to turn the ball away.
And they should have done much better three minutes later when Ismael Blanco caught Kenneth out of position but he shot wide from the edge of the area.
United then gave themselves a lifeline on 78 minutes when Paul Dixon's long ball fell for Daly inside the box and he prodded it past goalkeeper Sebastian Saja at full stretch.
Swanson had a chance to send United through on away goals with a drive from the edge of the box but it flew just wide.