Klopp’s team struggled in the first half and it was completely against the run of play when Sadio Mane put Liverpool ahead in first-half stoppage-time at St Mary’s.
Clinical finishing from Roberto Firmino after 71 minutes seemed to have secured the three points.
But a dreadful error by Liverpool goalkeeper Adrian gifted a goal to Danny Ings, who later miskicked from six yards when an equaliser seemed inevitable.
Reds boss Klopp had argued for a later start to the domestic season after his players’ busy international schedule during the summer.
And his team, who played extra-time in Istanbul before beating Chelsea in the European Super Cup on Wednesday, looked fatigued for long periods.
But after the final whistle at St Mary’s, Klopp insisted tiredness had not been a major factor in the uneven performance.
“Southampton are a good team, they defend well with good counter-attacks, good organisation,” he said.
“It was nothing to do with the game in midweek. If we had come here with two weeks’ preparation it would have been difficult.
“I liked the game a lot. I’m not allowed to think too much about Wednesday but of course I know it so I was really impressed with how fresh we looked, at the start especially and how good we were.”
Adrian, injured after colliding with a celebrating fan on Wednesday and playing with a swollen ankle, had to rush from goal to smother a shot from Che Adams, who also put a free header over the crossbar during Liverpool’s tepid opening.
And the Spanish goalkeeper was forced to make a reflex save when Japan defender Maya Yoshida beat Virgil Van Dijk to a corner kick and headed goalwards from three yards out.
But shortly before the interval, Mane took a short pass from James Milner, arguably the one Liverpool player who looked fresh throughout, and curled a beauty beyond the dive of Saints goalkeeper Angus Gunn.
Firmino scored the second after Mane had won the ball from a Southampton throw-in near their own corner flag.
“We controlled the game, we were fluent but yes, Southampton came up with some set pieces and long balls,” Klopp conceded.
“But Sadio scored a fantastic goal. That helped of course and in the second half we made more good steps.
“In minute 83 Adrian thought ‘I played really well so maybe I have to make the same mistake like Alisson Becker (last season) did and keep the game a little bit open!’.”
That was when Adrian collected a backpass from Van Dijk but then scuffed his attempted clearance straight at the right foot of Ings, the ball rebounding past him into the empty net.
If Ings’ late chance had gone in, Adrian might not have the light-hearted words from Klopp that he heard as he left the field.
“I said ‘You finally arrived, Ali did the same’,” Klopp said. “Obviously it’s a goalie thing at Liverpool. No problem as long as we win the games.
“Adrian had a swollen ankle so we played too many balls towards him but I was happy with everything he did.
“I don’t think the goal was because of that. He’s good with his feet and if you had seen his ankle after Wednesday then you would have said that even the pass to Ings for the goal was better than you would have expected.”