A howler from Dons keeper Jamie Langfield 11 minutes from time, in allowing Kris Commons' tame effort to squirm through his grasp, proved the difference between the sides.
The Pittodrie side looked comfortable enough as the champions struggled until Langfield made an horrendous mess of Commons' shot from distance at his near post after the Hoops midfielder had played a short corner with Joe Ledley.
Langfield, nicknamed Clangers, will not want to see his aberration again but Celtic will certainly have to do better in Finland on Wednesday night if they are to build on their 2-1 first-leg lead against HJK Helsinki in their Champions League third qualifying round tie.
It was a lacklustre display by Celtic, although goalkeeper Fraser Forster had little to do.
Celtic legend Sean Fallon, the 90-year-old former player and assistant manager, unfurled the championship flag before kick-off along with Hoops chairman Ian Bankier, but after that it all went flat.
Celtic skipper Scott Brown and James Forrest failed fitness tests and with defender Charlie Mulgrew suspended after being sent off in a pre-season game against Ajax, manager Neil Lennon was forced to reshuffle his pack.
In came stopper Thomas Rogne, midfielder Beram Kayal and striker Anthony Stokes.
Dons boss Craig Brown gave competitive debuts to former Celtic wide man Niall McGinn, Jonny Hayes and Gary Naysmith, the latter being lucky to escape an early booking for a lunge at Kayal.
In the 10th minute Celtic's stand-in skipper Georgios Samaras was caught with a flying boot from Aberdeen defender Ryan Jack but resumed after treatment to his bloodied face and with a new shirt.
The experienced visitors looked composed and confident for the most part, and capable of striking on the break.
However in the 21st minute Langfield was forced into a decent save from Commons' 25-yard drive, after the Scotland midfielder had combined with Kayal.
Moments later, Samaras curled a shot from the left-hand side of the box which went just wide of the far post.
There was little more of note until 10 minutes from the break when Commons set up Gary Hooper but the former Scunthorpe striker ballooned his effort from the edge of the box over the crossbar.
Celtic defender Victor Wanyama had to move sharply at the other end to stop an Aberdeen counter-attack which involved McGinn and ended with the Northern Irishman racing desperately to get on to a Jack pass inside the box.
Aberdeen had a half-chance soon after the restart when Hayes' corner from the right was punched only as far as Dons striker Scott Vernon, but from 12 yards he drove the ball high and wide.
Samaras, perhaps struggling from his early knock, was replaced by Kelvin Wilson in the 55th minute with Wanyama moving into midfield, but Celtic could not shrug off their sluggishness.
Commons again took it upon himself to inject some excitement into the match in the 63rd minute, firing in a low 25-yard drive which Langfield saved down at his left-hand post.
The home fans became increasingly exasperated as the out-of-sorts champions huffed and puffed with decreasing success.
Fans' favourite Paddy McCourt replaced Kayal with 20 minutes remaining, with the Dons bringing on Chris Clark for Naysmith minutes later.
Lennon's last throw of the dice came when he threw on striker Daryl Murphy for Stokes, and two minutes later the Hoops were in front when Langfield allowed Commons' speculative shot to squirm through his hands at the near post.
The relief among the home fans was palpable and they, like the Celtic players, were happy to see the game out before they start preparing for their Champions League encounter in Finland.
It will be a worrying tie for Celtic but it will be Langfield who will have trouble sleeping tonight.
