The Hoops were beaten 6-1 on aggregate by Legia Warsaw in the third qualifier but were reinstated after the Polish club were effectively thrown out by UEFA for fielding a player who should have been suspended in the second leg at Murrayfield.
Confident following their 1-1 draw in Slovenia last week, the Scottish champions were booed off at half-time following 45 minutes of struggle.
The introduction of Kris Commons at the start of the second half for Beram Kayal saw a vast improvement but Maribor skipper Tavares grabbed the crucial goal in the 75th minute, costing the Parkhead club around £20million in revenue but much more in reputation as the spotlight falls on boss Ronny Deila.
Celtic will go into the draw for the Europa League - but that will be scant consolation for the angry Hoops fans who made their feelings known at the final whistle.
Deila brought back his big guns as expected, although there was no place for last season's top scorer Commons.
The Norwegian made nine changes from the side that began the 1-0 defeat at Inverness on Saturday, with only stand-in skipper Charlie Mulgrew and defender Efe Ambrose remaining in the starting line-up.
Celtic had played their previous two home qualifiers this season at Murrayfield in Edinburgh as Parkhead was out of commission due to its use in the Commonwealth Games.
There was a big - but not capacity - crowd inside to see the home side make hard work of the first half after an encouraging start.
Maribor midfielder Agim Ibraimi's speculative shot from 30 yards sped just past Craig Gordon's left-hand post before Ambrose did well to block a close-range shot from Tavares after the Hoops defence had been cut open all too easily.
Gordon was called into action in the 28th minute when he parried a drive from defender Mitja Viler and was lucky to see the rebound hit Tavares and fall back into his arms.
Soon afterwards, a spell of Maribor pressure ended with midfielder Petar Stojanovic testing Gordon with a long-distance drive.
It was not going to plan for Deila's side and although they upped the tempo in the final five minutes of the half, visiting keeper Jasmin Handanovic saw little action.
The frustrated Hoops fans were happy see Commons come on for the start of the second-half and the substitution had an immediate impact.
Maribor found themselves pinned back in their own half and in the 55th minute they enjoyed some fortune when Celtic midfielder Callum McGregor, who had scored in the Hoops' three away European ties this season, hit the bar from Stokes' cutback.
The volume inside Celtic Park increased as the Slovenian side, stripped of their first-half composure, became increasingly desperate in their defending while offering little in attack.
Commons had the crowd on their feet in the 65th minute with a left-footed drive from 30 yards which just cleared the bar, before Adam Matthews replaced Jo Inge Berget.
However, there was a scare for Celtic moments later when Maribor worked the ball into their box in a rare attack with winger Dare Vrsic driving straight at Gordon from a tight angle.
In the 71st minute Derk Boerrigter replaced Mikael Lustig in another reshuffle before Commons screwed a shot wide from 16 yards.
But Parkhead was stunned when Maribor took the lead, the Slovenian side capitalising on Celtic's inability to clear their box when Tavares' shot appeared to take a deflection before looping over Gordon and into the net.
As the home side piled forward for the goal which would take the game into extra-time, Maribor broke and Gordon had to make a good save from Jean-Philippe Mendy's angled drive.
In the desperate last stages Ambrose headed a Stokes free-kick past the near post, then Tavares blasted over from 12 yards when he ought to have scored.
However, Celtic stopper Virgil van Dijk missed a great chance in added-time, shooting straight at the keeper from close range.