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The Reds were indebted to an outstanding goalkeeping performance from Alisson Becker before Elliott struck with his first touch off the bench in the 87th minute for the winner.
The result means Arne Slot's side are in a good position going into the second leg at Anfield in a fortnight having inflicted a first defeat on PSG since November.
Here the ECHO rates the Liverpool players after a very difficult but ultimately victorious night in France.
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Alisson Becker 10
Made two excellent stops to deny Kvaratskhelia, another to keep out Dembele and basically kept Liverpool in the game before the break.
Needed to be alert as Kvaratskhelia started to drift over to his flank second half and improved defensively.
Ibrahima Konate 6
Escaped a penalty and possible red for challenge on Barcola and done by Kvaratskhelia for one chance.
Slightly more composed as Liverpool sorted themselves out after break.
Virgil van Dijk 7
Stood up to PSG forwards, particularly at set-pieces, and set the example others started following second half.
Contributed to defensive effort second half, though.
Mohamed Salah 5
Bested by Mendes in terms of physicality far too often but didn’t get much service.
Did so well to set up the opener.
Curtis Jones (for Diaz 67) 8
Ability to keep hold of the ball for more than two seconds at least offered Liverpool some temporary respite and became big influence in closing stages.
Wataru Endo (for Gravenberch 79) 7
Defensive qualities were required as PSG continued to press.
Harvey Elliott (for Salah 86) 8
Introduced on to the right flank and scored the winner with his first touch