Tokmac Nguen latched on to a hopeful long punt forward to score Ferencvaros’s winner in the 75th minute, rounding off a miserable night for Neil Lennon’s Scottish champions.
Ferencvaros, managed by former Ukraine striker Serhiy Rebrov, go forward to the two-legged final qualifying round where they will bid to reach the group stage for the first time since 1995-96. Celtic drop into the Europa League qualifiers along with the continent’s also-rans.
Dutch club AZ Alkmaar needed an equaliser in the fifth minute of stoppage time to stay alive against Czech side Plzen before winning 3-1 after extra time.
Plzen led through David Limbersky’s 78th-minute goal but Teun Koopmeiners levelled from the penalty spot at the death.
Albert Gudmundsson put the hosts in front eight minutes into extra time before Koopmeiners sealed victory.