The Egyptian international replaced Darwin Nunez in the 68th minute and helped complete a 7-1 rout over the Scottish Premiership club.
Salah scored the three goals in six minutes and 12 seconds, taking the score from 3-1 to 6-1.
All three goals were assisted by Diogo Jota. It is the first time this has happened since March 2012 when Franck Ribery assisted three Mario Gomez goals for Bayern Munich against Basel.
French striker Bafetimbi Gomis previously held the record for the fastest treble in the Champions League.
He scored three times in eight minutes for Olympique Lyonnais against Dinamo Zagreb in the 2011-12 group stage.
“He played a different position,” he said. “A bit more inside, but of course, he is an outstanding player, as we all know, and today he showed it.”
The goals came at the right time for Salah’s confidence ahead of the club’s Premier League game against Manchester City on Sunday.
Klopp added: “This had nothing to do with the City game, but of course, it is better to get some confidence and some flow, but we now have to prove consistently.
“I don’t have to say it all the time, but I get asked all the time.”
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