Ambrose was at fault for both goals that the Bhoys conceded in the contest, but it was his misplaced back-pass that was the talk of the pundits after the match.
Celtic were leading their Turkish opponents 2-0 when Ambrose inexplicably tried to head the ball back to Craig Gordon, only to tee up Fernandao, who grateful accepted the gift and tucked the ball into the net.
That put Fenerbahce back into the match, and ultimately cost Celtic the three points.
And Poyet said: "Everything was on a plate for Celtic to win, but they made one massive mistake, absolutely massive. You cannot make it in a Sunday League game, yet alone a Europa League match."
The former Sunderland manager's analysis may have been scathing, but Hamann agreed with the harsh comments.
He added: "They were in total control, but for no reason Ambrose decides to loop a header in the air. It's a pathetic header."
If Celtic could have held out to gain the three points against Fener, then they would have been in a commanding position to qualify from their Europa League group.
However, a draw leaves their progression hanging in the balance, with Celtic needing to collect at least one win from their next two games against Molde to feel confident of making it through to the knockout stage.