It was unfortunate for the Dons as they had fought and scrapped for every ball and looked to have taken their place in the semi-finals.
Slack play by Stephen McManus in the third minute almost presented the home side with a dream start. Chris Maguire latched on to the loose ball before playing in strike partner Lee Miller who netted from close in, only for the goal to be chalked off for offside.
At the other end, Vennegoor of Hesselink met Paul Hartley's free-kick to head just over the bar in the eighth minute.
Aberdeen had a major let-off in the 23rd minute. Shunsuke Nakamura's cross was met by Scott McDonald whose first-time effort crashed off the upright.
From the rebound, Nakamura played the ball back across goal only for Dons keeper Derek Soutar to brilliantly turn away Vennegoor of Hesselink's header.
Celtic should have taken the lead five minutes before the interval when Darren O'Dea's cross found the head of Vennegoor of Hesselink. However, the Dutchman's effort came back off the face of the post and McDonald could only screw his effort wide of the gaping goal from six yards.
Aberdeen started the second half purposely and Alexander Diamond headed Scott Severin's 48th minute cross wide.
Three minutes later Aiden McGeady swept past Alan Maybury and played in a great cross to the far post where Vennegoor of Hesselink could only head the ball over from a good position.
Minutes later the visitors passed up an even better opportunity. McGeady played Nakamura in on goal but Soutar got down bravely to turn the ball away with Maybury on hand to clear the danger.
Dons keeper Soutar produced a sublime save to stop an Evander Sno thunderbolt on 62 minutes and then held McDonald's effort from the rebound.
Aberdeen almost took the lead in the 66th minute. Jeffrey de Visscher's corner found the head of Diamond whose downward effort was cleared off the line by Andreas Hinkel with Artur Boruc beaten.
The Celtic keeper was called into action again minutes later when a weak Hinkel clearance fell to Sone Aluko whose fierce drive was fantastically parried over by the big Poland international.
Aberdeen grabbed the vital opener in the 79th minute. Severin played the ball up to Miller who laid the ball in to de Visscher who smashed home from 14 yards.
Celtic's reply was immediate. Hartley picked the ball up in midfield before striding forward to crash in a great shot which ricocheted off the foot of the post.
Aberdeen could have been in the semi-final but for a tremendous double stop by Boruc, firstly from Severin's effort and then from the resulting corner he somehow blocked a goalbound Miller header.
With the last throw of the dice, Celtic pushed everyone forward in stoppage time for Nakamura's free-kick, won under dubious circumstances.
The ball was not properly cleared by the desperate Dons defence, allowing second-half substitute Georgios Samaras to play a delightful ball back into the box where Vennegoor of Hesselink gleefully slid his shot home.