With seven minutes remaining and his side up 2-0, the Spanish keeper botched a clearance attempt following a back-pass by kicking the ball directly at pressuring Saints striker Danny Ings who deflected the ball into the net.
Liverpool, though, held on to win their second match from two in the league and Klopp was in a playful mood about the incident afterward.
"Yes: 'You finally arrived, welcome!' [Alisson] did the same," Klopp said when asked what he told Adrian after the match. "Obviously it's a goalie thing at Liverpool, no problem with that as long as we win the games. All good."
Adrian took over as Liverpool's No. 1 last week after an injury to regular keeper Alisson, whose early days at Anfield after signing with the club last summer were marked by unforced errors as well.
Before Alisson's arrival, Liverpool also endured high-profile gaffes from Loris Karius in their 2018 Champions League final loss to Real Madrid in Kiev.
In his defence, Adrian was not playing at 100% on Saturday. He is still recovering from an ankle injury suffered when a supporter slid into him during the celebrations following Liverpool's UEFA Super Cup victory over Chelsea on Wednesday.
"Adrian had a swollen ankle and we played too many balls back to him in that period," Klopp added. "I was happy with everything he did today, all the saves, all that stuff. The other players have to then feel more the responsibility for the build-up and cannot give all the balls back to him and hope the pain-killers still help or whatever.
"I don't think the goal was because of that, but a few other balls were. He is completely good with his feet. If you would have seen his ankle on Thursday after we left the plane then you would say even the pass before the [Southampton] goal was better than you would have expected! It is all fine."
Klopp added that Adrian nearly did not play in the match at St. Mary's.
"He had a clearing session this morning at 10 o'clock and then the message flew to my iPhone and said he is fine," Klopp said. "If the message would have been he is not then I would have had to make a decision [not to play him], but until then it was all good. We were waiting for that. He said at breakfast he would be fine, but we had to test it and that's what we did."