Robson was on target for Celtic but the visitors could not hold on
A late own goal from Gary Caldwell saw Celtic crash out of Europe after the visitors had taken the lead in Aalborg.
Shunsuke Nakamura shot over from the edge of the penalty box and Georgios Samaras twice made a mess of close-range chances in the first half.
Barry Robson powered in a great header to put Celtic in front on 53 minutes.
But Caca's long-range strike was deflected in and Glenn Loovens' attempted clearance ricocheted in off Caldwell to give the Danes victory.
The Scottish Premier League leaders looked to have ended their miserable Champions League run on the road when Robson scored but it is Aalborg who qualify for the Uefa Cup, while Manchester United and Villarreal progress from Group E.
And Celtic had numerous chances to overrun a poor home side, who failed to threaten until the closing stages.
Indeed, Samaras and Nakamura passed up fantastic opportunities to score for the visitors inside the first 10 minutes.
The Greek striker mistimed his lunging bid to connect with a wonderful cross from Nakamura right in front of goal and the Japanese star ballooned a shot over the crossbar when one-on-one with goalkeeper Karim Zaza.
The Danes struggled to make any early impression in attack but Artur Boruc had to make a smart save to push away a flashing header from Michael Jackobsen, who was left unattended at a corner on 35 minutes.
The tenacious Scott Brown robbed Steve Olfers to set up a promising break deep in the Aalborg half but poor passes from Nakamura and then Samaras allowed the outnumbered home defence to clear up.
Samaras was left with his head in his hands again when he glanced a Nakamura free-kick wide from six yards out.
Aalborg's Thomas Enevoldsen blazed a rising shot over the top soon after the interval and Celtic were claiming a penalty when Scott McDonald tumbled between two defenders and was clipped by the diving Zaza.
Any sense of injustice was soon forgotten when Robson stole across his marker to meet Wilson's cross and steer a strong header low into the bottom corner of the net from 15 yards.
Caldwell, impressive in a holding midfield role, ditched his defensive duties to dance into the Aalborg penalty but the angle for the strike was tight and Zaza managed to beat out the Scotland international's stinging effort.
With the home team pushing players forward, spaces were opening up for Celtic and Nakamura fizzed an angled shot wide from 20 yards.
However, luck deserted Gordon Strachan's side 20 minutes from time as parity was restored from Caca's speculative dig from long distance.
The Brazilian was given too much room to line his sights but the ball took a huge deflection off Stephen McManus and looped over the despairing Boruc.
Both sides threw caution to the wind for the final 15 minutes and Celtic lost out in cruel circumstances as a low cross from Andreas Johansson was cut out by Loovens but was diverted back into his own net by the knee of Caldwell.
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