Arne Slot's much-changed side looked to be easing into the last four as goals from Darwin Nunez and Harvey Elliott inside the first 32 minutes put the holders in control at St Mary's Stadium on Wednesday night.
Yet the Saints, under the interim charge of Simon Rusk following the departure of Russell Martin, fought back in the second half.
Cameron Archer replied with a fine strike just before the hour and Mateus Fernandes felt he should have had a penalty in the closing minutes, but the Reds survived a nervy finish.
Uruguayan Darwin Nunez opened the scoring for Liverpool at Southampton. (AP PHOTO)
In London, Gabriel Jesus scored a hat-trick — including a crucial second goal from an offside position — as Arsenal beat Crystal Palace 3-2 to advance.
Replays showed that Jesus was offside for his goal that made it 2-1 in the 73rd minute at Emirates Stadium, but there is no video review until the semis of the competition. The Brazil striker, who had equalised in the 54th, completed his first treble for Arsenal in the 81st to give his team a two-goal cushion.
Former Gunners striker Eddie Nketiah ensured a nervy finish by heading home a cross in the 85th to reduce the deficit.
Jesus had previously only scored one goal in 20 appearances in all competitions this season — and that was also in the League Cup at second-tier Preston North End in October.
Two-time winners Arsenal last reached the final four in the 2021/22 season.
Gabriel Jesus scores one of his three goals in Arsenal's home win against Crystal Palace. (AP PHOTO)
At St. James' Park, Sandro Tonali's double helped Newcastle United secure their progress with a 3-1 win over Brentford.
Tonali put the Magpies ahead with a brilliant first-time strike and the Italy midfielder bagged his brace just before halftime from Anthony Gordon's corner.
Fabian Schar extended the lead after the break with a tidy tap-in before Yoane Wissa earned a consolation for the Bees in added time.
Victory sends Newcastle one step closer to ending their 69-year wait for a major domestic trophy.
Tottenham host Manchester United in the fourth quarter-final on Thursday.
With PA.
