Southampton will take a one goal advantage to Anfield in two weeks' time as they look to reach their first major cup final in 14 years.
Nathan Redmond popped up with the only goal of the game, although Southampton will undoubtedly feel that they should have scored more.
The Englishman alone should have had three, with Liverpool fortunate to be only one goal behind in the tie following an uninspired performance.
Liverpool dominated possession, but created next to nothing as Saints frustrated the Reds for the second time at home this season despite the visitors being in rampant form this year.
Saints set up to soak up pressure and break, and looked solid throughout the first half. After a frenetic opening that saw Redmond nearly get to a clever, dinked pass inside the Liverpool box, the home side settled back into their defensive mode.
Liverpool looked to be targeting Cedric Soares, with long balls over to his right back position. Saints were organised though, with Captain Virgil van Dijk commanding the area well.
Dusan Tadic hit an early cross straight to Loris Karius, and Ryan Bertrand intercepted a pass, ran 60 yards with Dejan Lovren muscling him off the ball well before the 20 minute mark.
Redmond looked to have wasted a fine opportunity in the 19th minute, with Karius saving after the ball fell to the Englishman in the box. Without a goal in 25 games, it seemed as though his dire run would continue. It didn't.
Klavan's error and Nathaniel Clyne's lack of concentration let Jay Rodriguez in, who fed Redmond with the forward slotting home with ease.
Ex-Saint Adam Lallana was Liverpool's only real bright spark in the first 45 minutes, dancing through a couple of challenges to earn a corner on one occasion before cleverly heading the ball into the onrushing Roberto Firmino's path, who forced a smart save from Fraser Forster.
Southampton came out of their shell following the goal, and began to pressure Liverpool higher up the pitch. On the occasions where the visitors looked like breaking, the midfield trio of Oriol Romeu, Steven Davis and Jordy Clasie were quick to mop up. The organisation was clear to see, even if Jay Rodriguez was isolated on his own up top at times.
Redmond should have doubled the lead just before half time, following great work down Southampton's left flank. Bertrand mugged Adam Lallana, with Tadic's cross falling to Redmond in the box. The goal was his to aim at, but he chose to hit another straight to a relieved Karius, a poor effort with his left foot. Liverpool escaped to the interval just the one goal behind.
The second half saw yet more Liverpool domination of the ball, but they didn't threaten the Southampton goal. In fact, it was Southampton who started the second half better, with Cedric earning an early free kick off James Milner just after the restart. Saints continued to press in spells, earning free kicks high up the pitch before settling back into a rigid formation.
Rodriguezcame into his own in the second half, linking up well with his teammates and showing good intelligence. Daniel Sturridge had a shot easily saved before the hour mark, and Lallana continued to create half chances but nothing more than that.
Philippe Coutinho replaced Georginio Wijnaldum on the hour mark, and he had an impact on the game but it was not enough to force an opening. Instead, Southampton should have put several more past Liverpool.
A fantastic break saw Redmond feed Cedric in the box, with the Portugal defender lashing a shot into the side netting when he should have either crossed it or at least hit the target. Ten minutes before time, a smart back heel from Rodriguez looked like it would lead to another chance, but Klavan managed to get a touch to the ball.
Substitute Shane Long came on and caused havoc with his pace, racing past Lucas before toeing the ball into Redmond's path. The Englishman chipped the ball over the keeper with Lovren and the bar combining to keep it out somehow.
Liverpool offered very little, Coutinho and Sturridge blazing long range efforts well over late on, yet Saints saw the tie out with ease.
While Southampton will be pleased to have something to defend at Anfield on the 25th, they will be kicking themselves that it wasn't more, such was their dominance over the gilt-edged chances to score.
Next up is Norwich, with Claude Puel's men having stopped the rot following three straight losses. Shout out to Maya Yoshida, who continues to impress at centre back - he will be needed with Jose Fonte having elected to effectively end his Saints career after handing in a transfer request.
Saints: Forster; Cedric, van Dijk (C), Yoshida, Bertrand; Romeu, Davis (Hojbjerg 73), Clasie (Ward-Prowse 82); Redmond, Tadic, Rodriguez (Long 82)
Subs: Lewis, Long, Ward-Prowse, Hojbjerg, Stephens, Sims,
Liverpool: Karius; Clyne, Lovren, Klavan, Milner (C); Lucas, Can, Wijnaldum (Coutinho 61); Lallana, Firmino (Origi 73), Sturridge
Subs: Mignolet, Gomez, Moreno, Stewart, Coutinho, Woodburn, Origi