Despite settling for a goalless draw in an uninspiring last 16, second leg at Anfield, Liverpool eased past Porto 5-0 on aggregate thanks to a comprehensive win in Portugal three weeks ago.
“It feels really good,” said Klopp, who is only in his second full season in charge.
“I think this year we belong there to be honest, it should not be a big surprise
“The next round will be difficult, seven other very good teams, maybe four of them are from England, which doesn’t make it easier.
“I think we will have a chance for sure to go to the semis. We’ll take it step-by-step and tonight it is cool.”
With the job done in Portugal, Klopp could even afford the luxury of starting top-scorer Mohamed Salah and club-record signing Virgil van Dijk on the bench with one eye on Saturday’s vital visit to Manchester United in the Premier League.
“Now we can prepare for the Manchester game. I’m looking forward to it,” added Klopp.
Liverpool have made a habit of fast starts in Europe this season, but took time to warm to their task with the German having made five changes in all from the side that beat Newcastle at the weekend.
It took 17 minutes for the hosts to threaten when Joe Gomez’s cross was acrobatically flicked towards goal by Mane, but looped just over the bar.
Mane was inches away from his fourth goal of the tie on the half-hour mark when he controlled James Milner’s lofted pass on his chest before smashing the inside of Iker Casillas’s far post.
Dejan Lovren then headed narrowly over as Liverpool threatened to raise the pulse of an understandably more subdued Anfield than is customary on a European night.
“Probably if you draw away and then win at home it feels maybe different,” continued Klopp.
“But the maximum you can achieve is the next round and we did that in a comfortable way.”
Casillas was given a standing ovation by the Liverpool fans in the Kop end as he took to the field after the break on his record 167th and possible farewell appearance in the Champions League.
“I think he has more (appearances) than our whole squad,” said Klopp, who embraced Casillas at full-time.
“I don’t say the number of years but I think he obviously wants to play a bit more.”
However, that was one of few moments that got the home fans off their feet as Liverpool settled on their substantial lead.
Roberto Firmino had one clear chance to add to his seven Champions League goals this season when his shot was blocked by Porto captain Felipe before the Brazilian was protected by Klopp for the weekend as Danny Ings was introduced for the final half hour.
Salah was given 15 minutes to find the winner when he replaced Mane.
The Egyptian had one opportunity to extend his scoring streak to eight straight games, but Casillas smothered his low effort.
And the Spanish veteran made his best stop of the night two minutes from time to deny Ings and hold onto his clean sheet.