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Fenerbahce countered in the 10th minute through Ozan Tufan, who sent an incisive cross into the Hoops' box that had to be scrambled to safety for a corner, which came to nothing.
Despite this, Celtic continued to push forward and went on to open the scoring in the 28th minute, as a Commons corner fell to Griffiths, who bundled home after several attempts.
Just four minutes later Ronny Deila's charges doubled their lead, with Commons edging into space inside the box, latching on to a Forrest ball and slamming a shot into the top left corner.
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It would get even worse for the hosts as, just four minutes after the restart, the 28-year-old Brazilian leapt highest to get onto Nani's corner and thump a header past Craig Gordon to nullify the visitors' deficit.
The pendulum had well and truly swung the way of Fenerbahce, as they piled forward and laid siege to the Hoops' goalmouth through Hasan Ali Kaldirim and Diego, but Gordon was equal to both their efforts.
Celtic, however, had a sterling chance to retake the lead in the 70th minute, with Commons fashioning a long-range chance which Fabiano repelled convincingly.
Despite the hosts' best endeavours, they could not get themselves back in front, and the final whistle signalled a sharing of the points between themselves and Fenerbahce.
Celtic sit second in Group A following the result, while Fenerbahce slip to the foot of the table despite their point.