The Parkhead side dominated the Group E clash at the Energi Nord Arena from the first whistle and there was no surprise when midfielder Barry Robson put them ahead in the 52nd minute with a diving header.
Gordon Strachan's side looked to be on their way to winning their first away tie in the group stages of the competition at the 18th attempt, but in the 72nd minute, and out of the blue, Caca drew his side level with a drive from 25 yards which deflected off skipper Stephen McManus before looping over Artur Boruc.
And when Gary Caldwell put through his own goal three minutes from time the visitors' misery was complete and they do not even have the consolation of a UEFA Cup place to look forward to in the New Year.
It was quite a night in Denmark but one which the Scottish champions will want to forget in a hurry.
As the teams came out, the Aalborg PA system blared out the Rangers anthem 'Simply The Best' to add some heat to a bitterly cold evening.
The temperature perhaps contributed to the low-key start but Celtic were first to threaten and in the sixth minute Shunsuke Nakamura curled in a wonderful pass from the left which evaded the Aalborg defence, but also Georgios Samaras who arrived just too late.
Three minutes later the talented Japan international passed up a great chance to forge his team ahead.
Scott McDonald's first-time ball to the right put Nakamura clear through on goal but from just inside the box he knocked the ball over Aalborg keeper Karim Zaza and the bar.
In the 22nd minute Aalborg eventually offered something of a threat when Thomas Enevoldsen headed a Martin Pedersen free-kick past Boruc's right-hand post.
The Danish club appeared strangely nervous and subdued with striker Jeppe Curth offered no support from his team-mates as he struggled to get any change from Glenn Loovens and McManus.
As the game drifted towards the break the home side struggled to impose themselves.
However, in the 35th minute Caca had a shot from 25 yards which was deflected for a corner and when the ball came in from Martin Pedersen, Boruc pulled off a fine save under the bar from Michael Jakobsen's header.
But four minutes from half-time Celtic missed another gilt-edged chance after Caca had fouled Mark Wilson outside the box on the left-hand side.
Nakamura flighted the free-kick into the six-yard box to the unmarked Samaras who headed past the far post when it looked like he should have at least hit the target.
When the second half restarted Celtic were soon in control again.
In the 51st minute McDonald was sent tumbling in the box by Zaza but Austrian referee Konrad Plautz ignored desperate and apparently valid claims from the Hoops players and supporters for a penalty.
But a minute later there was plenty for the small band of Celtic fans to cheer about when Robson put them ahead with a wonderful diving header.
The Parkhead side again moved up the park with ease and when Wilson's cross came in from the left, the former Dundee United midfielder stretched to guide the ball past Zaza and into the corner of the net.
For the first time in the game the Danish side showed signs of urgency and in the 63rd minute Enevoldsen had a close-range header but his effort failed to trouble Boruc.
Celtic, however, sensed there was a second goal there for the taking and Caldwell skipped his way through the Aalborg defence before sending in an angled shot from 10 yards which Zaza blocked at his near post.
Tension grew as the game stretched further and in the 71st minute Nakamura broke before firing in a 30-yard drive which bounced past Zaza's left-hand post.
But a minute later the visitors were stunned when Caca picked up a Kasper Bogelund pass and sent in a drive from 25 yards which took a deflection off McManus before going over Boruc.
The game changed on that incident.
Celtic were on the back foot and there was further disaster when Loovens thrashed a clearance from Andreas Johansson off Caldwell and into his own net from close range.
Celtic's search for that elusive away win in Europe continues while the post-mortem begins.
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