Klopp fielded basically a second-choice lineup at St. Mary's Stadium, with the game coming only three days after Liverpool's energy-sapping FA Cup final against Chelsea that went 120 minutes before the Reds won a penalty shootout.
Despite the slew of changes, Liverpool dominated a passive Southampton team with nothing left to play for this season but still fell behind in the 13th minute when Nathan Redmond cut inside from the left and sent in a shot that deflected off James Milner and into the far corner.
Liverpool's players kept their composure and replied 14 minutes later when Diogo Jota laid the ball off to Takumi Minamino, who displayed quick feet before firing a rising shot inside Alex McCarthy's near post. The Japan forward was on loan at Southampton last season and hadn't started a league game since then.
The rest of the match saw Liverpool camped inside Southampton's half but only managing to create half-chances. The atmosphere was starting to get tense when Liverpool was gifted what proved to be the winning goal, after a corner by Kostas Tsimikas was glanced on at the near post by Southampton forward Mohamed Elyounoussi.
The ball flew across the area, skimmed off the head of Southampton's Kyle Walker-Peters, hit the forehead of Matip and rebounded into the far corner beyond McCarthy's despairing dive.
Liverpool also came from behind in its last two league games, a 2-1 win against Villa and a 1-1 draw with Tottenham.
