Senegal international Mane agreed the fresh deal – a reported two-year extension running until June 2023 - on Friday (NZDT) having hit six Premier League goals in Liverpool's unbeaten start to the season.
Mane, 26, makes up one of the most fearsome frontlines in world football along with Mohamed Salah and Roberto Firmino, the trio firing the Reds to the UEFA Champions League final last term.
And a thrilled Klopp feels there could still be more to come from Mane, who has hit 40 goals in 89 appearances for the club since joining from Southampton for £30million in 2016.
"He is a such an important member of our team and squad," Klopp said. "He radiates joy and I think this is reflected in his performances and impact on the pitch.
"The only criticism I could ever have of Sadio is that maybe at times he is the only one not to see just how good he is - but as an attacking player he is the complete package.
"He made a big impact when he first arrived, but if anything he's got better and better each season. His trajectory has only been upwards in terms of his level and his contribution. I know his teammates will be as delighted as we are and also the supporters.
"Maybe the other thing which is lost on some who don't know Sadio is just how ambitious and hungry he is to achieve things and be successful. He is more driven than I think many will be aware of.
"He wants to do it with us and by committing the best years of his career to LFC I know we have someone who will give their absolute everything... as a manager you can ask for no more."
Mane, who was linked with joining Real Madrid prior to the shock departure of Zinedine Zidane, follows Salah and Firmino in signing new Liverpool contracts this year.
"This is a big boost for everyone and another demonstration of how the players we have at this club are committed, not just to our present but our future also," Klopp added.
"I can't think of a single club anywhere in Europe that wouldn't want a player like Sadio available to them, so the fact he wants to remain with us says something about where we are currently."