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The Reds were by far the better side throughout and built up a commanding two-goal lead, but looked to have thrown it away after shipping once before the break and again in the closing stages.
Thomas Meunier pulled one back for the Ligue 1 champions, though Edinson Cavani was offside in the build-up and not spotted.
And a late equaliser from Kylian Mbappe, previously anonymous, had most of the ground feeling extremely frustrated after a dominant performance.
But Firmino’s close control and fantastic finish won the match late on in deserved fashion.
For PSG boss Thomas Tuchel, it was a case of lightning strikes twice; he was in charge of Dortmund when the Reds also prevailed with a last-ditch winner in the Europa League shortly after Klopp took over at Liverpool.
This was a massive first step for the Reds as they look to come through a difficult group, with Napoli away next on the agenda—and the Italians themselves could only draw 0-0 away to Red Star Belgrade on Tuesday.