Footbal Sensation: Maribor in Champions League
Maribor captain Marcos Tavares scored the winning goal by chipping a loose ball from eight metres out over Celtic goalkeeper Craig Gordon, after a tense 74 minutes in which Maribor failed to capitalise on its chances.
Ante Šimundža's men entered the match needing an away goal to have a shot, after the clubs played out a 1-1 draw in Maribor last Wednesday.
They would control the play early on with some creative midfield play and defensive pressure as Celtic struggled to get out of the gate.
Maribor's Macedonian forward Agim Ibraimi would first threaten the home goal in the 16th minute, but his strike from around 20 metres went slightly wide.
Celtic caused problems for Maribor through the air in the first match and defender Virgil van Dijk came close from a corner in the 22nd minute.
But it was Maribor which continued to create the bulk of the chances and in the 29th minute midfielder Mitja Viler found space in the Celtic box only to see his shot saved by Celtic goalkeeper Craig Gordon.
Gordon, Celtic's best man on the pitch, was called to make another save only moments later, when Aleš Mertelj had a go from 20 metres out.
It would then be Maribor goalkeeper Jasmin Handanović's turn to put on the heroics as Ronny Deila's outfit found their game at last late in the first and early in the second.
The crossbar saved Maribor as Callum McGregor tried to slot home a half volley from seven metres in the 56th minute.
The lucky escape would breathe new life into the Slovenian club, who again found purpose in the midfield. They created their first chance of the second half in the 66th minute, when midfielder Dare Vršič broke free down the left side of the box only to blast the shot directly at Gordon.
Unable to find a way past Gordon, Maribor were looking at their third defeat in the Champions League playoffs in as many years, before Tavares swooped on a failed clearance from Celtic in the box to slot it over the keeper.
Maribor should have put the match away in regular time but wasted three excellent opportunities on goal, including a four-on-two break which ended with substitute Dejan Bohar fumbling the ball away.
This gave Celtic one last chance to force extra time, as Virgil latched on to a pass early in added time with a strong strike only to see Handanović produce a wonderful stretch save.
After having to settle for the UEFA Europa League in the past two years, Maribor is returning to Europe's top flight for the first time since 1999, their maiden appearance in the competition.
Šimundža, who was part of the 1999 Maribor team, becomes the first Slovenian to reach the Champions League both as a player and coach.