Ronny Deila's men were unable to take advantage of playing the second leg at home as their European lifeline proved short-lived, Tavaras with the decisive second-half goalCeltic were eliminated from the Champions League for the second time this season after losing 1-0 at home to Maribor and 2-1 on aggregate.
Ronny Deila's men had been given a stay of execution in the previous round thanks to Legia Warsaw fielding an ineligible player but, after drawing in the first game, the Scottish champions failed to exploit home advantage against the Slovenian side.
Marcos Tavares got the decisive goal, swinging his right leg at the ball to loop it over Craig Gordon 75 minutes into proceedings.
Deila's side had failed to properly clear their lines and paid the price for poor defending.
Celtic had a chance to force extra time right at the death when the ball fell to Virgil van Dijk from just six yards out but the Dutchman aimed his effort straight at Jasmin Handanovic.
The Scottish side had been largely comfortable throughout much of the match but failed to make their greater control count, with large parts of the game simply listless and lacking in quality.