Salah's new deal will reportedly make him the club's highest paid player of all-time on £350 000 a week until 2025.
The 30-year-old had entered the final year of his previous contract with talks stalling over his wage demands.
But tying Salah down to a new deal was key to Liverpool's hopes of continued success under Klopp.
In five years at Anfield, the Egypt forward has scored 156 goals in 254 appearances.
"This is a special treat for our supporters to enjoy their weekend even more. I'm sure there will be some celebrations for this news tonight," Klopp told Liverpool's website.
"It's the best decision for us and best decision for him. He belongs with us I think. This is his club now.
"I have no doubt Mo's best years are still to come. And that's saying something, because the first five seasons here have been the stuff of legend."
Since joining from Roma in 2017, Salah has become a fundamental part of reestablishing Klopp's men as serial contenders for domestic and European success.
He scored in the 2019 Champions League final victory over Tottenham and was the club's top scorer when they won the Premier League for the first time in 30 years in 2020.
Salah has more trophies in his sights after finally putting pen to paper.
"I feel great and excited to win trophies with the club. It's a happy day for everyone," he said.
"It takes a little bit of time, I think, to renew, but now everything is done so we just need to focus on what's next."
"I think you can see in the last five or six years the team was always going (upwards)," Salah added.
"Last season we were close to winning four, but unfortunately in the last week of the season we lost two trophies.
"I think we are in a good position to fight for everything. We have new signings as well.
"We just need to keep working hard, have a good vision, be positive and go for everything again."
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