The Georgia star cut in from the left flank midway through the second half and, with typically quick feet, skipped past a defender and the goalkeeper before stroking in his team-leading seventh goal of the competition.
Desire Doue put the defending champions ahead in the 11th minute in Paris on Wednesday with a deflected effort. Ballon d'Or winner Ousmane Dembele spurned three chances at Parc des Princes.
"We had chances to score more," Kvaratskhelia said. "But I think it's OK, 2-0 is good but we have to stay concentrated."
When the two teams met last season in the round of 16, PSG advanced on penalties after losing 1-0 at home and winning 1-0 at Anfield.
The task seems less difficult this time around with Liverpool facing a trophyless season. However, the six-time champions overcame a 3-0 first-leg loss against Barcelona in the 2019 semifinals before winning 4-0 at home.
Liverpool manager Arne Slot's team was reeling from a crushing 4-0 FA Cup defeat to Manchester City but was better led this time by captain Virgil van Dijk.
Playing for the first time with five defenders this season, Slot explained his decision to be so cautious.
"They have pace from everywhere all over the pitch, no matter where you look," he said.
"They have so many weapons (and) the times when we tried to play high and aggressive they ripped us apart."
Atletico meanwhile triumphed 2-0 at Barcelona's Nou Camp, with Julian Alvarez converting a free-kick in the 45th minute awarded for a last-man foul by Pau Cubarsi for which the Barcelona defender was sent off.
Lamine Yamal and Marcus Rashford tried to rally Barcelona and kept the pressure on Atletico despite being undermanned.
But that fightback effort was undone when substitute striker Alexander Sorloth shrugged off a challenge by Gerard Martín as he moved to the near post and tapped in a cross from Matteo Ruggeri to double the lead with 20 minutes to play.
"We are heading home happy for the win, but we still have a long way to go. There are 90 minutes ahead," Atletico forward Antoine Griezmann said.
It was the fourth match in 2026 between the two, with Atletico winning their Copa del Rey tie 4-3 on aggregate while Barcelona won their league match in Madrid 2-1 on Saturday.
"We gave it our all, but it wasn't to be," Barcelona substitute defender Ronald Araujo said.
"But this team can turn this around. I am convinced of it. It will be tough at their home, but this team has the character and talent to come back."
On Tuesday, Bayern Munich won 2-1 at record winners Real Madrid and Arsenal prevailed 1-0 in Lisbon at Sporting in the other Champions League quarterfinals.
The four return legs will be played next week.
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