The visitors had just taken a 3-0 lead in the third qualifying round second-leg tie at the Antonis Papadopoulos stadium in Larnaca when play was abandoned in the 82nd minute.
The Cyprus Mail said on its website that Anorthosis fans invaded the pitch and pelted police with stones.
UEFA said in a statement: "The UEFA Control and Disciplinary Body has confirmed Anorthosis Famagusta FC's elimination from the UEFA Europa League after their third qualifying round second leg against FC Dila Gori was abandoned due to crowd disturbances.
"The Cypriot club were 3-0 down at home against their Georgian opponents when the game was called off. The match has been declared forfeit, but as the score was 3-0 to Dila Gori when it was abandoned, this result stands.
"In addition, Anorthosis have been ordered to play their next three UEFA club competition home games behind closed doors and fined 50,000 euros. An appeal can be lodged against this decision within three days of the dispatch of the reasoned decision.
"Dila will now play Portugal's CS Marítimo in the play-off round."