Ronny Deila’s side were cruising the Group A fixture through goals from Leigh Griffiths and Kris Commons until a horror mistake by the Nigeria defender two minutes from the break allowed substitute Fernandao to pull a goal back.
The Brazilian striker, on in the 40th minute for the injured Volkan Sen, headed in his second goal three minutes into the second half as Ronny Deila’s home side appeared still to be shell-shocked.
The visitors took control but the Parkhead side eventually composed themselves and deserved to finish with a point – it takes them to two points after two games – but will hope Ambrose’s error will not prove crucial by the time the group is completed.
It was a big night for 18-year-old left-back Kieran Tierney who replaced the suspended Emilio Izaguirre to make his first European start, while midfielder Stefan Johansen came in for Tom Rogic.
There was talk about a rift between Fenerbahce’s summer signing Robin van Persie and coach Vitor Pereira in the build-up to the game but the 32-year-old Dutchman started, supported from wide by former Manchester United team-mate Nani.
James Forrest looked in the mood as Celtic drove forward with purpose in the early stages and two goals in quick succession had the Istanbul outfit rattled.
A Commons corner from the right in the 28th minute was helped on by the head of Mikael Lustig and although Fenerbahce goalkeeper Fabiano blocked Griffiths’ first shot from three yards, the Scotland striker fired the rebound high into the net.
Parkhead erupted for the second time in the 32nd minute when Commons’ left-footed drive from 10 yards out, after taking a Forrest cut-back, gave Fabiano no chance.
The visitors looked bereft of belief but a remarkable Ambrose mistake in the 43rd minute gave them hope.
Fernandao could not believe his luck when the defender was hopelessly short with a back-header, allowing the big striker to knock the ball past Hoops goalkeeper Craig Gordon.
Fenerbahce replaced Ozan Tufan with Brazilian midfielder Diego for the start of the second half and within three minutes they were level when Fernandao bulleted in a Nani corner with his head.
The Hoops looked as if they could unravel completely.
The visitors’ two-goal striker almost got a third in the 52nd minute when he lofted the ball over the on-rushing Gordon but also over the bar, before the Celtic goalkeeper made a fine save from Hasan Ali Kaldirim.
Celtic rallied and in the 70th minute Fabiano made a fine save from a Commons thunderbolt of a strike and was then lucky after fumbling a tepid drive from Lustig.
The Hoops fans were pushing their side on in the three added minutes but there was to be no late escape for them, or Ambrose.
Gordon, Lustig,Boyata, Ambrose, Tierney (Blackett 82), Commons, Brown, Bitton, Forrest (Armstrong 90), Johansen (Rogic 82), Griffiths.
Fabiano,Ozbayrakli, Kjaer, Bruno Alves, Kaldirim, Sen (Fernandao 40), Tufan (Diego 46),Topal, Meireles (Ba 78), Nani, van Persie.
Kenn Hansen (Denmark).