Koeman made two inspired substitutions at the interval and was rewarded handsomely as Sadio Mane, one of his new men, scored twice.
Liverpool had looked on course for three points that would have put them firmly into the Champions League reckoning when they led at the interval through Philippe Coutinho and Daniel Sturridge – and it could have been more.
But Saints were transformed and could even afford to miss a penalty three minutes into the second half, before Mane and Graziano Pelle won them the match.
Koeman described the game as ‘must-win’ as it keeps his team on course for a possible return to the Europa League. “We got six points from Stoke and Liverpool which makes it possible,” he said. “That’s good because that means you are fighting for something until the end of the season.
“The team performance in the second half was perfect. It wasn’t good enough in the first half but Liverpool let us off lightly after 45 minutes. We punished them for that and I know that feeling because we lost many games when we had to kill the game and we didn’t.”
Klopp believes that his team can still make the top four but admitted that it will be difficult. “There are still 27 points to get,” he said. “If we can get all of them there is a chance. If not it will be difficult. But we don’t have to dream of things like this, we have to play football and win games.”
His team could have been a goal down in the first 10 minutes when former Saint Dejan Lovren pulled down Shane Long but referee Roger East was unimpressed and Liverpool made the most of their escape when Coutinho shot past the possibly unsighted Fraser Forster from 25 yards.
Sturridge made it 2-0 from a pass by Divock Origi and more quick breaks could have resulted in further goals for Joe Allen but he hit one shot too close to Forster and saw another hit the net only to be disallowed because Mamadou Sakho was offside and interfering with play, having jumped over the ball as it went past Forster.
A repeat of Liverpool’s 6-1 Capital One Cup win here looked on, so the turnaround in the second half was a shock. While Mane and Victor Wanyama upped Saints’ game, Martin Skrtel, replacing Lovren at the interval, had a nightmare.
First he pulled Pelle’s shirt to concede a penalty and was rescued when Simon Mignolet saved Mane’s kick but after that he never got close to Pelle. The Italian striker sent Mane in to pull one back after 64 minutes then equalised in the 83rd minute, beating Skrtel in the air, exchanging passes with Long and hitting a left-foot rocket past Mignolet’s right hand.
Three minutes later, Skrtel made a hash of a routine defensive header, Pelle pounced and played in Mane and he shot home. Koeman ran along the touchline in delight, although he confessed that “It is the first time and I think the last time, I am still tired.”
Klopp’s mood was darker. “I’m not that good a loser so I don’t feel brilliant as you can imagine,” he said. “But I’m long enough in football to know this can happen. I’m not angry, I am frustrated that we got nothing for a very good first half.
“We made football mistakes and we can learn from them and we will. But for today it’s too late because we lost.”