An early Shane Long strike left the Reds fearing that they would remain behind Manchester City in the Premier League title race.
But Klopp's men came firing back with goals from Naby Keita, Mohamed Salah and Jordan Henderson to seal three points at St. Mary's in a 3-1 victory.
The beaming manager complimented his team and their mental strength as they returned to the top of the table.
“Mentality, huh? It’s brilliant,” the German coach said after the game. "We came here and we knew Southampton is a really different side since Ralph [Hassenhutl] is in and we knew it.
“Everybody expects and we needed three points and we knew how difficult it was, and you saw it.
“The setup was 4-5-1 and you play against nine. You saw each ball we played into the box we played into the box and couldn’t finish off there was a counterattack because they were in better positions.
“They scored that goal, deserved, but after that we adjusted or adapted to the situation, we were there, won balls back, could play in better spaces, had our moments, and scored a goal. Then it was game on.”
There were also words of encouragement for Salah after the Egypt hitman scored his 50th goal for Liverpool in record time to end an eight-game drought, though his manager says he never doubted the forward’s quality.
“It was never in doubt about him being a great player from my side.
“It was goal no. 50 for Liverpool – that’s an incredible number.
“As a striker you have chances, each striker is a human being as well, maybe you have better moments and less good moments.
“If it’s a big thing we don’t really care about it but it’s a nice moment for his first goal in a while, a really good moment.”
Klopp added that there was no controversy after his team appeared not to celebrate Naby Keita’s equalising goal – his first for the club.
“I don’t know exactly why [they didn’t celebrate], it’s for sure nothing to do with Naby!
“They were really happy for him in the dressing room, he had multiple hugs.
“[It was] 1-1 – because we wanted to win so they boys obviously decided not to celebrate it like crazy because there was still a job to do.
“A fantastic moment for the first goal.”