Marco Reus made one for Henrikh Mkhitaryan and scored another in the first half and while it was over to Robert Lewandowski in the second half, with the striker scoring twice to keep Zenit at bay despite goals from Oleg Shatov and Hulk, the latter a penalty.
"We were close to the optimum," Klopp said. "The team were sensational, it was a great game. My side did a great job of closing down the space and creating chances."
Sebastian Kehl also credited the team's pressing for the win.
"It was very important that we scored two goals in the opening five minutes as it steered the game in the right direction right from the start," he told the team's website. "Zenit were rattled and we were able to play our game.
"With our pressing game, we won back possession, created space and a lot of chances, and eventually scored the goals."