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Substitute Kris Commons came off the bench to earn Ronny Deila’s side a point away from home.
Celtic were their own worst enemy again and were left kicking themselves after it Craig Gordon’s error.
The game’s opening goal prevented Celtic winning their first Europa League game this season.
Chances were few and far between in the first half for both sides, but Lazar Markovic drove at the Hoops defence and forced Gordon into an error, leaving the keeper in no man’s land.
Gordon came to Celtic’s rescue a number of times earlier in the first-half, including a fingertip save onto the post and a terrific double save.
Diego was sent off after lashing out at Stefan Johansen and gave the Hoops a chance to get back into the game.
Deila’s decision to take Scott Allan paid dividends instantly, as Commons’ diving header gave the away side a chance in the game.
Fenerbahce held on for a draw in the end, as Celtic couldn’t find the winning goal against 10-men.
Here are the player ratings from this evening’s game in Istanbul.