Liverpool had demolished Southampton 6-1 in the quarter-finals of this competition last season at St Mary's Stadium, but a repeat meeting came in sharp contrast as the hosts defended superbly throughout.
Virgil van Dijk again displayed his class with a strong, composed performance and how Jurgen Klopp will have wished for such a resolute presence in his own defence when Ragnar Klavan's error allowed Redmond to grab a potentially precious goal.
And as ex-Red Luis Suarez was scoring his 100th goal in 120 Barcelona games since leaving Anfield, Klopp's attack also came up short - a half-paced Daniel Sturridge failing to impress on a rare start that is unlikely to be repeated on this evidence.
Philippe Coutinho emerging from the bench just past the hour after almost two months out with an ankle injury will have warmed visiting hearts as Klopp prepares to unleash the Brazilian at Manchester United in Sunday's return to Premier League action.
That trip to Old Trafford becomes a key fixture for Liverpool, who have fallen into a mini stumble after being held 0-0 by Plymouth Argyle in the FA Cup on Sunday.
The Saints had scored first before being picked apart in last season's quarter-final tie and they again struck first, despite Liverpool dominating possession.
Redmond came into the game without a goal in 15 outings and nerves were evident when he took a touch from Cedric Soares' cross, allowing Loris Karius time to smother his shot, but the England Under-21 man atoned a minute later.
Klavan miscued when attempting to cut out Dusan Tadic's pass to Jay Rodriguez, who squared for Redmond to calmly finish across Karius and into the net.
Fraser Forster had earlier been called into action by Roberto Firmino's snapshot, but that was as much as the Reds mustered in 45 minutes, with Coutinho sent to warm up before half-time - perhaps a threat to those on the pitch from Klopp if nothing else - with Karius' razor-sharp reflexes denying Redmond a second and keeping the visitors in it just before the break.
Former Saint Sadio Mane appears to be a sore loss from Liverpool's front line, which lacked its usual zip without the Senegalese on Africa Cup of Nations duty, though Coutinho embarked on a couple of mazy runs after he replaced Georginio Wijnaldum.
As Liverpool's stodgy play continued to let them down, Southampton ought to have doubled their lead when Redmond led a lightning-quick counter-attack from a visiting corner, only for Cedric to slash wide from 12 yards when Rodriguez was supporting him infield.
On another day, Redmond would have single-handedly ended the tie inside 90 minutes and he clipped the crossbar with a late chip - his three misses something to pore over if Liverpool do turn the tables at Anfield in two weeks' time.