Liverpool qualified for the Champions League group stages in midweek with a 6-3 aggregate win over Bundesliga side Hoffenheim, and will be up against Sevilla, Spartak Moscow and Maribor.
Liverpool were 3-0 up within 21 minutes at Anfield in their 4-2 second-leg victory, and Klopp admits it was some of the best football he's ever been a part of.
In an exclusive interview with Sky Sports ahead of the clash against Arsenal on Super Sunday, Klopp said: "I think it was really deserved, and that's important, we would have taken it by luck, but that was the best way to do it [deserving it]. It was not perfect, but it was really good.
"The first half an hour against Hoffenheim was really some of the best football I've been a part of! So I feel really good in this building, working all together, with players and coaches, all the staff around, really good.
"But of course we have to show it outside also, and it's our job, and I really hope we can reach something special, for this special club."
Liverpool host Arsenal on Super Sunday, live on Sky Sports Premier League from 3.30pm, and Klopp has taken seven points from a possible nine against the Gunners during his time at Anfield.
While Borussia Dortmund manager in 2013, Klopp said he prefers "heavy metal football" to Arsenal's "orchestra" style, and when asked about the comment, the German admitted the comparison was wide of the mark.
"It's hard, in times of television and internet and the world wide web, nobody forgets things like this.
"No! [It's not a great quote]. It wasn't even right! I never thought that anybody would remind me about it one more time after that! Four years ago and you're still coming up with this!"
Klopp even admitted that Arsenal do change their style when they play against Liverpool, and says they can be devastating after winning the second ball.
"I think Arsenal change style always when they play us, you watch the games back and it is like this. When we played at Anfield, the 3-3 [in January 2016], there were a lot of long balls to Giroud, for his chest.
"Maybe [because of the way we press], we don't press all the time, so you can play! It's high intensity, high quality, they're really good.
"They have a quite clear build-up, winning the second ball, and then they play football, and they're all involved. Mesut Ozil, Sanchez, whoever, Lacazette, Giroud, and it's really difficult to step in, but that's a part of the game, we have to work hard again.
You can't play a little bit of football and hope to get anything from Arsenal, they are too strong, I really respect [them]."
Liverpool were impressive against top-six opposition last season, finishing top of the 'mini-league', but Klopp joked they did not win a trophy for that feat.
"We didn't celebrate the little championship! We were first in the top six league, but unfortunately just fourth in the whole league! Haha! We should change the rules!"