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Hansi Flick's men were second best for periods of the match at the Estadio da Luz in Lisbon, but Coman made the difference to deny his former club a historic first Champions League title.
His well-placed header ensured Bayern emulated its treble-winning side coached by Jupp Heynckes in the 2012-2013 campaign.
Given the exploits of both sides en route to the final, the first half was somewhat lacking the deadly touch many might have expected in the final third, with Neymar, Kylian Mbappe and Robert Lewandowski all spurning decent chances.
Coman, a PSG academy product, broke the deadlock in the 59th minute and that caused the contest to open up significantly.
But, while the Parisians might feel they were incorrectly denied a penalty in an incident involving Mbappe and Joshua Kimmich, Bayern held on to seal its sixth European crown.