The Dons were under constant pressure from the home side, but their defence was immense as they closed down to hold on to their single-goal lead in the closing minutes of the match.
Celtic had the better of the play and possession in the first half, asking plenty of questions of the Aberdeen defence. But the visitors' backline was not shaken by the attack and goalkeeper Derek Soutar had few saves to make.
Jan Vennegoor of Hesselink headed over the bar in the opening minutes after good work by Lee Naylor, while Massimo Donati's long-range effort was easily gathered by the keeper.
Karim Touzani made an excellent challenge to dispossess Shunsuke Nakamura as he closed in on goal.
Celtic's best chance of the first half came five minutes before the break, as Scott McDonald sent a left-foot shot just wide of the upright with Soutar beaten.
But it was the Dons who nearly took the lead as half-time approached as Stephen McManus sent a slack pass back to Artur Boruc.
Rickie Foster dashed in to intercept and Boruc made an excellent save to help his side leave the pitch on level terms.
Aberdeen were almost immediately under pressure again as the match resumed. Zander Diamond suffered a cut eye and a bloody nose after a series of aerial clashes against Vennegoor of Hesselink and the visitors played for time with ten men.
Celtic tried to capitalise on their man advantage, but Aiden McGeady shanked an effort wide and then blasted a 30-yard effort just over the bar.
Totally against the run of play Aberdeen took the lead on 69 minutes. Vennegoor of Hesselink failed to clear an Andrew Considine header and Mackie was on hand to score just his second goal of the season.
The Dons faithful must have feared a repeat of the first quarter-final match when Celtic grabbed a late equaliser, as the Glasgow giants threw everything forward to try and net the equaliser.
But Soutar made a fantastic save from a Nakamura free-kick and Considine deflected a Vennegoor of Hesselink shot over the bar.
Celtic even brought goalkeeper Boruc up in the dying seconds as they won a corner, but it is Aberdeen who will line up against Queen of the South at Hampden Park next month after a famous victory.