It was a competitive first 45 minutes between the two sides but it was Sadio Mane's header on the stroke of the half-hour mark that saw the Reds lead 1-0 at half-time.
A top-four place still remains a tall order, but the result sees Liverpool close the gap to fourth-placed Leicester City to six points, with a game in hand, with the Reds still in the hunt.
Klopp was forced to field an inexperienced centre-back pairing of Nathaniel Phillips and Rhys Williams, with Ozan Kabak missing due to a knock. Diogo Jota started in attack as Roberto Firmino dropped to the bench.
The Reds had much of the ball inside the opening ten minutes, without particularly threatening the visitors. However, Sadio Mane had the first real opportunity of the match to make it 1-0 when he controlled well and cut inside from the left, only for his shot to blaze over the crossbar.
Saints began to show their threat in glimpses as the half wore on, with Nathan Redmond causing problems down Liverpool's right and Kyle Walker-Peters finding pockets of space between Mane and left-back Andy Robertson.
But Mohamed Salah had a glorious chance to put the Reds ahead as the ball fell to Egyptian from close range, but his shot went straight at Fraser Forster's feet before Nathaniel Phillips headed wide from the ensuing corner.
A well-worked move between Mane and Jota saw the latter run through on goal before getting his shot off, but Forster was up to the mark as he palmed his shot behind.
Liverpool were knocking on the door as Salah and Gini Wijnaldum then went close in close succession, with Forster keeping Salah's curling effort out before Liverpool's captain on the night struck the crossbar with a powerful header from a corner.
However, Southampton had the pace to hurt Liverpool and they nearly took the lead after a quick break but for Alisson, who stopped Che Adams' close-range effort before dealing with Redmond's strike from the edge of the box.
The Reds took the lead less than a minute later as Salah found Mane with an excellent lofted ball, with Mane heading home past Forster to end his recent barren run to make it 1-0 to Liverpool.
Salah could - and should - have doubled the advantage after a fantastic pass by Jota. But the Reds forward took too long before getting his shot off, which was blocked after Southampton got a host of bodies back to defend.
Nathan Tella had a chance to level things just before the break after a Redmond cross, but his close-range header, fortunately, went straight into Alisson's arms.
Jota could have made it 2-0 at the start of the second half with a shot from a tight angle being turned behind by Forster for a corner, from which Phillips headed wayward.
Another well-worked move by Redmond saw Che Adams find Michael Obafemi with a threaded pass, but Alisson was quick off the mark to deny the substitute a goal.
But a shocking lack of composure from Alisson a few minutes later saw him gift the ball to Che Adams but the Reds goalkeeper got down well to save his blushes after the Saints forward got a strike quickly out of his feet.
Mane thought he'd made it 2-0 to seal the three points for Liverpool but the Senegalese was correctly adjudged offside from Trent Alexander-Arnold's lofted ball from the left.
Despite a nervy few minutes, Liverpool eventually found a crucial second to seal the win, with Thiago scoring his first goal for the club with a brilliant finish into the bottom corner.
Liverpool will likely need to win their remaining four fixtures to be in with a chance for a top-four place, but it's a result that sees them still in with a chance, with Leicester City just six points adrift having played a game more.
MOTM: Nathaniel Phillips
Next Fixture: Manchester United (A) - 13 May, 20:15 BST