Following a cagey opening to the game, Paul Hartley almost put the SPL champions in front.
Kenny Miller played the ball through to the former Hearts man who saw his strike well-saved by Jamie Langfield and, as the rebound fell to the Scotland midfielder, he failed to get his second effort on target.
Derek Riordan then blasted wide from a Shunsuke Nakamura reverse pass as Celtic created the better of the early chances.
Vennegoor of Hesselink broke the deadlock after 33 minutes knocking home the rebound after Jamie Langfield could only parry Riordan's effort.
Aberdeen almost equalised minutes later when Steve Lovell saw an effort well-blocked by Artur Boruc.
The Dons then had an even better chance to level when they were awarded a spot-kick when Lovell was tugged back inside the area by John Kennedy.
However, Boruc made a marvellous stop, diving to his left to save Jamie Smith's penalty.
The Dons eventually grabbed a deserved leveller four minutes before half-time when Darren Mackie netted from 16 yards, driving the ball low across Boruc and into the bottom corner of the net.
The Pittodrie outfit could have taken the lead in first-half stoppage time but Boruc produced another save to deny Lovell.
But it was Vennegoor of Hesselink who scored the winning goal at the start of the second half, netting with a deft flick from Hartley's cut-back.
Aberdeen had chances to level in the second half, with Zander Diamond going closest with a header that hit the crossbar.
The battle for the final UEFA Cup spot will now go down to the final day of the season after the Dons saw their lead over Hearts, who won the Edinburgh derby against Hibs, cut to a single point.