Zenit St Petersburg defeated Dortmund 2-1 in Germany on Wednesday but the home side held on for a 5-4 aggregative triumph in the last 16.
While Klopp was disappointed by the way his team defended their nervousness in the first half, the 46-year-old was thrilled to have reached the Champions League's last eight for the second year in a row.
"My lads deserve to be in the quarter-finals because they always fight like wolves," Klopp said.
"When you see who else is in the quarter-finals of the Champions League, then this is the creme de la creme.
"We stand for the second consecutive year in this round and I'm totally happy."
Zenit took the lead in the 16th minute thanks to Hulk's long-range strike before Sebastian Kehl's header dragged Dortmund level seven minutes before half-time.
Salomon Rondon struck the winner for the visitors after 72 minutes, but the Russian Premier League club could not find another goal to increase the pressure on their hosts.
Klopp hailed Zenit as 'a great team' but was frustrated how his side failed to utilise their Champions League experience from three consecutive campaigns to control the match effectively.
"You could sense a bit of tension and a degree of nervousness," Dortmund's boss said.
"After we scored the equaliser the match wasn't decided but we were calmer on the ball. We've got the experience of 20 or so Champions League matches, including a final, but we could have played a lot calmer in the first half."
Dortmund striker Robert Lewandowski warned his team-mates they will have to improve if they are to advance past the quarter-finals.
"This was not a perfect game by us. I do not know what it was," he said.
"Of course we are very happy about the quarter-finals.
"There is now waiting definitely an absolute top team for us. To beat them, we have to give 110 per cent and play much better than today."
