The striker hooked his shot from eight yards over Craig Gordon as the Celtic defence struggled to clear the ball from a cross from the left flank.
Celtic’s defeat means they will not be in the group stages of the competition for the first time in three years.
Maribor survived a couple of early scares with Anthony Stokes inches away from connecting with an Emilio Izaguirre delivery in the opening exchanges.
Marko Suler then cleared the ball an inch past his own post after Jo Inge Berget got to the goal-line and sent a cut-back through a crowd of defenders.
At the other end Agim Ibraimi picked the ball up 30 yards from goal and shot a foot wide with Gordon at full-stretch after 15 minutes.
Maribor had looked vulnerable from set-pieces in the first-leg last week and Celtic came close with a Stokes corner which Virgil van Dijk headed narrowly wide of the post.
But Celtic were tested again when Mitja Viler stepped inside Efe Ambrose and drilled a shot from 16 yards which Gordon parried against Tavares, before the keeper smothered the rebound.
The home side stepped up the pace after the break and were on the end of some tasty challenges from Maribor which almost led to a breakthrough when Ambrose clipped a Stokes free-kick wide of the post.
Callum McGregor then hit the crossbar from six yards after Stokes created space to go past a defender and cut the ball back to the youngster.
Kris Commons sends a dipping shot from 30 yards just over the Maribor crossbar and later dragged a low drive wide of the far post.
With 10 minutes remaining, Ambrose met a Stokes free-kick at the front post but powered his header wide with the keeper beaten.
Tavares then shot wide from 16 yards with the goal at his mercy in the dying moments as Celtic were stretched at the back.
And then there was real drama deep in stoppage-time when Van Dijk drifted off his marker to take a McGregor chip on the chest but his first-time volley was well saved by Jasmin Handanovic.