Kazakhstan's 30,000 Astana Arena was the venue for the Hoops for Tuesday night's play-off round first-leg tie, with the teams battling out the clash on an artificial pitch.
Neil Lennon made three changes from Saturday's 2-0 Scottish Premiership win at Aberdeen with Mikael Lustig, Virgil Van Dijk and James Forrest all coming into the line-up.
The Scottish champions enjoyed the better of the opening stages with debutant Van Dijk testing goal keeper Aleksandr Mokin with a powerful header from a Charlie Mulgrew corner.
However, it was Shakhtyor who took the lead against the run of play on the 12-minute mark. Celtic failed to deal with a long throw-in from the left which was flicked on for Andrei Finonchenko to prod the ball past Fraser Forster from close-range.
Kris Commons rattled the bar with a thunderous drive in the 19th minute with the ball bouncing down onto the line, but the Scots struggled at times with the route-one approach taken by a home side who often looked limited in other aspects of their game.
Having already found success from long throw-ins, the hosts opted to regularly utilise Gediminas Vicius' powerful deliveries, causing Celtic all sorts of problems at the back throughout the game.
Lennon's side endured a majorly frustrating first half but they did come close to equalising just before the break when Giorgios Samaras headed Mulgrew's inch-perfect free-kick wide.
Celtic started the second half brightly, with Mulgrew finding himself denied by Mokin before the goalkeeper then pulled off the save of the game to keep out Forrest's header after Emilio Izaguirre's perfect delivery.
But the Hoops were made to pay the price for their defensive frailties once again when Sergei Khizhnickenko headed past Forster after the ball was deflected into his path.
Steven Mouyokolo did well to block a goal-bound shot from Khizhnichenko but when the ball was crossed back in to the box by Vicius, the striker steered home a header from 10 yards.
The Glasgow giants were stunned and the final minutes passed in something of a blur for the visitors, with Mulgrew heading over from six yards in the final minute.
As the game drew on, Lennon decided to bring summer signing Amido Balde off the bench as they pushed for that elusive away goal, but they were unable to find an opening in what was a frustrating evening for them.
After getting past Cliftonville and Elfsborg in their previous two qualifiers, the Parkhead side now need to pull something special out of the bag next week, although in front of their own fans that is far from unlikely.
The two sides meet again next Wednesday at Celtic Park, with the Hoops having it all to do if they are to play Champions League football this season.