Having gone ahead through a clinical Kylian Mbappe strike, PSG found itself trailing after goals from Andre Silva and Nordi Mukiele either side of half-time.
It set the stage for a Messi masterclass with the forward scoring twice to put the French giant back in front.
The first came after some slick play with Mbappe and the second was trademark Messi. The 34-year-old held his nerve to coolly dispatch a Panenka penalty in the 74th minute to ultimately win the game.
But there was still time for more drama with PSG being awarded another penalty in the 94th minute of the match when Achraf Hakimi was bundled over in the box.
It seemed nailed on that Messi would step up to complete what would have been his first hat-trick since moving to the club but, instead, it was Mbappe who took it.
The French forward blazed his effort clear of the crossbar and Messi was left with his head in his hands.
“It’s normal, it’s respect,” Mbappe told reporters after being asked why Messi took the first penalty.
“He’s (Messi) the best player in the world, it’s a privilege he plays with us, I’ve always said it.
“There’s a penalty, he takes it, period. For the second, he said, ‘Take it.'”
‘Important for us to win’
Despite a slow start to his life at PSG, Messi seems to be clicking into gear with his new team.
His link-up play with the excellent Mbappe was yet again a highlight of the latest win, which looked to be in some doubt against the impressive Leipzig who pressed high throughout the match.
“It was a tough game, but we’re very happy to have picked up the three points,” said PSG boss Mauricio Pochettino.
“It was important for us to win and go top of the group. It was difficult against a very good team, but we deserved the win.”
PSG now sits atop of Group A with Leipzig, still without a win in this season’s competition, bottom.
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