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 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 finding their best form.
"Today we were close to playing at our optimum level," the head coach said. "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 on March 19.
Manchester United eyeing surprise move for AC Milan star for whom tension in Italy is growing: report
‘Even though Kevin Keegan was my boyhood hero, we never really saw eye-to-eye. That was an underwhelming time for me – I felt on the outside’: Newcastle cult hero reveals disappointment over how things went under King Kev
