Following a 1-1 draw in Slovenia last week, the Glasgow club needed only to draw the game 0-0 to progress into the group stage for the third year in a row.
As it turned out, Tavares' strike on 75 minutes sent them spinning into the Europa League, losing out 2-1 on aggregate.
The Hoops got off to a frantic start, putting Maribor on the back foot from the off.
However, despite their early pressure, a Virgil van Dijk header whistling by the post was all they had to show for it.
At the other end Mitja Viler drew a good save from Celtic goalkeeper as the Slovenians began to press going into the break.
After the interval Celtic kicked into gear. The introduction of Kris Commons seemed to inspire the home side as Callum McGregor rattled the crossbar from an Anthony Stokes cutback.
Ronny Deila's men were to regret missing that opportunity, however, as the visitors took the lead with 15 minutes to go, Tavares capitalising on a wayward Emilio Izaguirre attempted clearance which sprang back into the box.
Efe Ambrose headed just wide five minutes later as Celtic pushed for the goal to take the tie to extra-time, while van Dijk fired straight at Jasmin Handanovic from five yards with seconds to go.