Liverpool entered the second leg against the Portuguese side holding a 5-0 advantage from leg one and largely coasted to a draw as Klopp trotted out a side with five changes.
But the Reds boss was not concerned with what was largely a lacklustre display, and instead put his focus on the side reaching a level the club hadn’t seen in Europe in nearly a decade.
“It feels good, to be honest” he said of reaching the last eight. “It feels really good.
“I cannot feel what people feel they wait so long for it, but it feels good and I know about it, of course.
“The last eight is cool. We’re a little bit satisfied and it’s already good. I think it was time we showed up again.
“There is good development in Liverpool in the moment, it’s a nice moment, but if you don’t show up in a competition like the Champions League then nobody really will realise it.
“I’m happy for all the people involved that we have made that step. So it’s done, it’s good, now let’s carry on.”
Klopp admits that he never had a timeline in mind for Liverpool’s return among the European elite, but claims this year’s Liverpool side are deserving of their place among the continent’s best.
“I never had a timescale for it in my mind and thought ‘then we have to be there and then we have to be there’. I don’t want to make it smaller than it is, but I don’t want to make it bigger than it is.
“I think this year we belong there, to be honest - it should not be a big surprise.”
Liverpool face a wait to learn their fate in the quarter-finals, with the draw set to take place on March 16.
Klopp knows that nothing will come easy for his side and, with as many as four other Premier League sides possibly in the draw, that the possibility of facing an English rival exists.
But the German boss believes his side can push on no matter who they draw in the next round.
“The next round will be very difficult, I think that’s clear,” he said.
“There are seven other very good teams that will be involved. Maybe four of them are then from England and that doesn’t make it easier, to be honest, but I think we’ll have a chance, for sure, to go to the semis and that’s the target.
“But you’re right, take it step by step and tonight it’s cool, absolutely. This year we had the first opportunity to do it and now we are in it, so I think we are in time.”
Liverpool return to Premier League action on Saturday with a crunch clash against Manchester United.