Robert Lewandowski struck twice as die Schwarzgelben bounced back from a disappointing defeat against Hamburg with a fine European triumph in Russia
Goals from Marco Reus and Henrikh Mkhitaryan in the opening five minutes put the Bundesliga side in control of the last-16 first-leg tie and, although they let Zenit back into the game, last year's beaten finalists remained ruthless in front of goal.
Oleg Shatov and Hulk found the back of the net for the hosts, the latter with a penalty, but Robert Lewandowski struck in between. The Poland international then grabbed his second of the night to hand the visitors a strong advantage ahead of next month's return leg.
Klopp praised the attitude of his side for bouncing back from a league defeat to Hamburg on Saturday, and felt that they were closing to produce their very best form.
"Today we were close to playing at our optimum level," the head coach told Sky. "The team were sensational, it was a great game.
"My side did a great job of closing down the space and carving out chances.
"We also knew that Zenit were very dangerous especially when Hulk was on the ball.
"We let Hamburg have the game the way they wanted, but it was clear we'd show a reaction tonight."
The two sides face each other again at Signal Iduna Park on March 19.