Jordon Ibe added his name to the scoresheet as Liverpool emphatically overcame the early blow of Sadio Mane putting the hosts ahead inside a minute to emphasise their resurgence since Klopp arrived at the start of October.
Klopp's side, who have now won seven of their last eight games in all competitions, will face Stoke City in the semi-finals while Premier League leaders Manchester City will play Everton in the other last-four clash after Wednesday's draw.
While Liverpool's improvement under Klopp, especially away from home, has sparked a wave of renewed belief among their fans, they were second best and a long way off the pace as Southampton grabbed an early stranglehold on the tie.
A little over 40 seconds were on the clock when Mane stole a march on Liverpool left back Alberto Moreno and arrived at the far post to get his head to a cross from Ryan Bertrand and bury the ball past Adam Bogdan.
Liverpool, however, seem to have developed a steelier constitution under Klopp and turned the game around thanks to an inspired five minutes from Sturridge.
Having returned from a long injury absence as a substitute against Swansea City on Sunday, he shook off the remaining ring rust to draw Liverpool level on 25 minutes.
The England striker miscontrolled a long, arcing pass from Joe Allen, but collected the ball, beat the final defender and drilled a left-footed finish into the far corner of the net.
His second, four minutes later, was a much easier finish and owed a lot to a brilliant pass with the outside of his foot from Emre Can, with Sturridge arriving late in the area to prod the ball home.
Liverpool's third came on the stroke of halftime as a corner from the right found its way to Alberto Moreno, who drilled the ball low and hard towards goal, allowing Origi to stick out his boot and get the faintest of touches to direct it into the net.
His second goal was far more emphatic. He raced on to a ball forward from Ibe and lashed it into the roof of the net from the edge of the area to effectively finish the tie.
There was further gloom, however, for Southampton as second-half substitute Ibe drilled home from 20 metres to make it five before Origi completed his hat-trick with an 86th-minute header.
Original source: Liverpool hit the Saints for six