Southampton, who had also won 1-0 in the first leg, had the best chances in the first half although Liverpool enjoyed the bulk of possession.
First Tadic and then Davis could have put the Saints ahead as Liverpool could not turn their dominance into creation.
The game changed after the break as Liverpool pushed the visitors back into their own half and the Saints hardly found any chance to counter-attack.
Sturridge however was wasteful in front of goal on the two occasions he found himself in a position to open the score.
Meanwhile Southampton goalkeeper Forster managed to save himself from embarrassment after almost giving away a goal to Liverpool following a long-range strike by Can.
Liverpool appealed for a penalty in the 90th minute and perhaps overlooked the great run by Sims that flew through the middle and rolled it over to Long who aimed for the bottom left-handed corner of Karius.