Tokmac Nguen's second-half goal sent the Hungarian side through and consigned the Scottish champion to a third season in a row without reaching the group stage.
A superb strike from David Siger put Ferencvaros ahead after only seven minutes with what was their only shot in the first half.
Celtic, which finished the game with 70 per cent of the possession and 28 shots to the visitor's seven, was level after 53 minutes thanks to a powerful effort from Ryan Christie.
Olivier Ntcham rattled the crossbar with a half-volley and Mohamed Elyounoussi had a goal disallowed for a marginal offside before Ferencvaros' sucker punch.
Nguen ran onto a long clearance, held off a tame challenge from Hatem Elhamed and squeezed a shot beneath Vasilios Barkas.
Celtic will now drop into UEFA Europa League qualifying, while Ferencvaros head into Tuesday's draw for the next round.