Driving rain on the south coast made conditions difficult for both sides early on, and it took 19 minutes for the first attempt to arrive, though Tyler Dibling’s effort for the hosts was comfortably dealt with by Caoimhin Kelleher.
Five minutes later, things were much less comfortable for Kelleher’s opposite number, Alex McCarthy, when a long ball forward from Jarell Quansah took a deflection and sat up kindly for Darwin Núñez, who slotted the ball past the goalkeeper, who slipped as he tried to set himself.
A deflection helped the Reds out for their second goal just after the half-hour mark, as good work on the left between Cody Gakpo and Wataru Endō set up Harvey Elliott on the edge of the area, and a nick off Ryan Manning helped take his strike past McCarthy.
The former England stopper saved face soon after with a brilliant stop to deny Alexis Mac Allister, but the record 10-time winners of the competition were firmly in control at the break.
Rusk’s inaugural HT team talk must have been inspiring, as the Saints made a good start to the second half and got their reward just before the hour mark, when the ball ricocheted kindly off Endō into the path of Cameron Archer, who cut in from the left and curled the ball past Kelleher to halve the deficit.
Just three minutes later, Archer almost completed the comeback, but this time a reaction stop from Kelleher denied him following Yukinari Sugawara’s cross.
Archer’s goal was Southampton’s 10th in four matches in the EFL Cup, but despite plenty of late pressure, they couldn’t force an equaliser, with the best late chance falling the visitors’ way, but Taylor Harwood-Bellis denied Federico Chiesa with a goalline clearance.
However, scoring at least twice for the 11th time in their last 12 games was enough for Liverpool to advance and go 20 matches unbeaten in all competitions, keeping them in the hunt on four fronts.