Looking to turn over a new leaf after their 3-0 Scottish League Cup humiliation at Rangers’ hands, Celtic dominated the ball early on, leading to opportunities for James Forrest and Camilo Durán.
But the Scottish champions didn’t take their chances, and were hit by a sucker punch after 20 minutes.
Bamidele Yusuf drove inside from the right, and his fierce left-foot cross-shot took a heavy deflection off Liam Scales as the ball flew into the bottom left-hand corner, giving the visitors a surprise lead.
And that lead nearly became something more substantial moments later, as Viljami Sinisalo smartly kept out a low shot from former Celtic winger Daniel Arzani.
It was a difficult watch at times for the Parkhead faithful, but after struggling to create clear-cut chances, Celtic dragged themselves level via a flowing move down their left.
Oliver Sørensen, Forrest and Kieran Tierney combined, and the latter crossed for Mika Baur to sweep home left-footed.
However, Celtic Park had barely finished celebrating when Ferencváros retook the lead with the final touch of the half.
Yusuf twisted his way down the right and found Kristoffer Zachariassen inside the six-yard box, allowing the Hungarian outfit’s captain and top scorer to apply an instinctive back-heeled finish to stun the home crowd.
Celtic brought on record signing Kasper Høgh at the break but it was ex-Celtic man Arzani who provided the decisive impact early in the second half to extend the visitors’ lead.
The Australian, who didn't score during an injury-hit spell in Glasgow, cut onto his right foot following a fast counter-attack before rifling a stunning effort into the top corner from 20 yards.
Martin O'Neill eventually called for the cavalry, including Benjamin Nygren and Sebastian Tounekti.
Nygren curled an effort inches wide while Tounekti tested Ádám Varga as Celtic tried to claw their way back into proceedings. Varga then gifted the ball to Nygren, who lifted the ball agonisingly wide from 25 yards with the goal gaping.
With the wounds of that UEFA Champions League qualification playoff collapse in Linz perhaps still too fresh, Celtic were ultimately taught a harsh lesson in finishing, despite some flashes of promise.
They must now move on quickly, with the first league Old Firm match of the season on Sunday lunchtime looming large. Meanwhile, Ferencváros will take a lot of encouragement from this, even though they’ve now kept just one clean sheet across their last 10 away fixtures in the UEL proper.
Flashscore Man of the Match: Dele (Ferencváros)