The hosts had an early chance to find the breakthrough when Mehmet Topal's low effort was tipped onto the post by Craig Gordon inside the first three minutes, while Nadir Ciftci's blocked effort was all that the Scottish side had to show for their early endeavours.
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Gordon beat Lazar Markovic to Gokhan Gonul's long ball, but he allowed the ball to squirm past him and the winger nipped in to convert a simple opportunity and put the Turkish side on the brink of qualification.
Celtic's defensive work had been resolute up until the goal, but Markovic was buoyed by his first Fener goal and began to cause increasing problems for the visitors' defence after the break.
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However, the home side were put under unnecessary pressure with 23 minutes remaining when a moment of madness from Diego saw them reduced to 10 men when the Brazilian was shown red for a petulant kick out at Stefan Johansen.
Celtic took advantage of their extra man within eight minutes to draw level, as substitute Commons powered home a header from his first touch when he met Mikael Lustig's curling cross to peg the hosts back with 15 minutes to play.
That leveller gave the Scottish Premiership side the late ascendancy and they had chances to snatch a first Europa League victory when James Forrest tried his luck and Callum McGregor curled inches over after exchanging passes with the winger.
However, Celtic had to settle for a single point as they ended their Europa League group campaign without a win in six matches, while Fenerbahce had to make do with second place in Group A.