The reigning European champions were forced to hold on for the three points at St Mary's Stadium after a mistake from the Spanish goalkeeper set up a nervy finish in the pair's Premier League clash, which the Reds emerged from as 2-1 victors to keep a spotless record for the season so far.
With a comfortable lead thanks to goals either side of half-time from Sadio Mane and Roberto Firmino, the former West Ham man gifted the hosts a lifeline when he misplaced a pass straight to the feet of Danny Ings inside the penalty area, allowing the former England international to bury it in the back of the net.
Adrian's gaffe bore a suspicious similarity to an error made by regular Liverpool keeper Alisson last season against Leicester , for whom the 32-year-old is currently deputising following an injury sustained by the latter.
Speaking after the match, Klopp however made light of the incident, quipping that the mistake means that he has now truly 'arrived' while backing him to bounce back.
"No it wasn't [out of our control]. If you give a goal like this, the other team is coming up and the other teams feels the intensity of the last week," the German told BBC Sport after the match.
"Southampton had one more big chance and if they score then we can't moan about that - but I think we deserve three points. I'm happy with the performance.
"We gave them a proper boost by that nice pass by Adrian. He has finally arrived because it is his third game and so it is normal for a Liverpool keeper to tick that box [by making a mistake]! Now we can carry on."
The Reds have suffered a number of goalkeeping snafus in recent years, with the infamous 2018 Champions League final display from Loris Karius the most memorable.
Former starter Simon Mignolet also had his moments of uneasiness, but the Belgian has since departed the club in a summer move to Club Brugge.
Alisson became the unquestioned starter at Anfield last season upon joining from Roma, as the Reds shipped just 22 goals in 38 Premier League games this past season following the Brazilian's £66 million ($84 million) move.
His injury however has presented a chance for Adrian, who arrived as a free agent only weeks ago, with the former Betis man etching his name into club history in midweek during Liverpool's Super Cup penalty shoot-out victory over Chelsea.