The veteran has penned a deal to 2022.
"He is our vice-captain and obviously one of the big leaders in our squad," Klopp told the club's website.
"He is just as important to us off the pitch in the dressing room as he is on the field, and that tells you everything as he is a big, big player for us. He is a big voice in the dressing room and someone all of the players look up to and respect."
He also stated: “One of the best examples of this is how he was on the morning of the Barcelona game at Anfield when he arrived at Melwood – as soon as he walked through the door that day, he was pushing everyone around him and motivating them and I loved it.
“And one of my favourite things about Millie, of which there are many, is he is genuinely still improving all the time as a player.
“He has an outstanding attitude: towards football, towards looking after himself and towards looking after others. He is an outstanding footballer and we all believe he still has a lot left to give, which is why this deal makes so much sense for all.
“He is an exceptional player, professional and person."