Bayern Munich took the lead in the first half when Kingsley Coman pounced after Dedryck Boyata’s hesitation.
The home side played valiantly and were rewarded by Callum McGregor’s equaliser after the break at Celtic Park.
The scoreline was only level for four minutes, with Martinez suffering a cut as he headed the winner.
With Paris St-Germain defeating Anderlecht 5-0 in the other Group B game, Celtic cannot finish in the top two positions, although they have an advantage over the Belgian side in the race for third and participation in the Europa League knockout stages.
Bayern were shorn of some of the key men in their defeat of Celtic a fortnight ago in Munich, with Robert Lewandowski, their goal-scorer in chief, and Thomas Muller, their rangy schemer, both injured.
Bayern’s manager Jupp Heynckes – with an eye on the Bundesliga meeting with Borussia Dortmund at the weekend – also relegated Mats Hummels, Thiago Alcantara and the hugely-influential Joshua Kimmich to the bench.
Celtic had not won a home group game in Europe in three years coming into the game, a stretch that comprised six losses and three draws – a run that does nothing to add to the legend of Celtic Park as a European fortress.
There was pressure on them to at least perform and take Bayern to the wire by making a proper game of it.
From early on you sensed they had it within them to do it. Their passing was crisp, the intensity high and their threat genuine. Bayern, by contrast, were way below their customary level.
Brendan Rodgers went with a 3-4-2-1 that became 5-4-1 when required. James Forrest and McGregor were whirling dervishes in attack and defence, buzzing about, breaking play up and then roaring forward on the attack.
Up front on his own, Moussa Dembele had a real presence.
Rodgers’ team should have been ahead as early as the third minute, when a lusty intervention from Kieran Tierney turned defence into attack.
Forrest took it up from there, dinking a cross to the back post where Stuart Armstrong looked sure to score, but the Scotland man side-footed wide of the post.