Their only goal came after Joel Matip turned into his own net from a cross from Serge Gnabry.
Liverpool then scored with headers from Van Dijk in the 69th minute and Mane in the 84th to book their spot in the last eight. "Important money-wise but also confidence-wise", Klopp told reporters. "We are back. There is no doubt we are back again". We are back on the landscape.
"That was part of the plan when we started the project".
"Scoring three here is really hard - actually we scored four because we scored the other one as well!" he told a post-match news conference.
The manager said victory at one of European football's powerhouses, following on from last season's appearance in the Champions League final, confirmed Liverpool's place among the elite in the game.
Liverpool join Barcelona, Man City, Man United, Ajax, Porto, Juventus and Tottenham in the last eight and will learn their opponent following Friday's UCL draw.
"We were playing at home but that wasn't our game today and that's why Liverpool beat us".
"It's a big step for us".
Image Liverpool FC via Getty Images
And in doing so, he set an all-time goalscoring record for the Anfield club! "I knew we had a chance and the boys made it happen".
James Milner echoed Klopp's praise of Mane's goal, saying the forward displayed "ridiculous" skill while scoring Liverpool's first goal.
"Sadio is on fire".
Sadio Mane's inspired spinning finish inside 26 minutes delivered just what the Reds were after and allowed them to seize control of a heavyweight encounter. "Hopefully that continues for the rest of the season". This club always has outstandingly strong football teams but we kept them away from our goal as well as possible. It's never easy, they're going to have spells on top.
"I'm not happy, I'm angry because we know we should do better but we didn't".
"We defended pretty well, didn't give them too much, and the front three were outstanding again".
Information from Reuters was used in this report.