Redmond's first-half strike was the difference at a packed-out St Mary's, bringing a timely end to Southampton's four-match winless run in all competitions on their first appearance in the last four of a major competition since 2003.
Claude Puel's men rallied manfully after a sluggish start against the side second in the Premier League, with Redmond capitalising on Ragnar Klavan's mistake to coolly slot home.
Unfortunately for Saints the England Under-21 international did not show the same cutting edge with his two other first-half opportunities, while Cedric's decision to shoot rather than cross after the break may return to haunt them.
Redmond saw an effort hit the underside of the bar as the hosts attempted to add a deserved second, but they will have to make do with just a one-goal advantage in the return leg at Anfield on January 25.
Southampton looked to be in for a bleak evening as they struggled in the opening stages, defending deeply and struggling to find an attacking outlet.
Liverpool were exposing the full-backs with ease as Puel's team started nervily, but Fraser Forster denied Roberto Firmino when the visitors eventually managed to turn that dominance into a clear-cut chance.
Jay Rodriguez was looking overwhelmed as Saints' focal point yet there was a flash of life in the 19th minute as Redmond collected a cross from the right and got away a shot that Loris Karius smothered well.
Redmond was caught out by the goalkeeper's reactions, but made amends within a minute as Klavan's inability to deal with a simple ball forwards cost his side.
Rodriguez was alert to capitalise on the hashed clearance, beating former team-mate Dejan Lovren with a deft pass to put Redmond through and this time the 22-year-old did not falter, keeping his composure to direct past Karius.
That sucker punch after such a comfortable start threw Liverpool off kilter, failing to attack with any intensity before half-time.
Southampton instead ended the opening period on top. Jordy Clasie saw an audacious attempt comfortably dealt with by Karius, who was a match for Redmond's close-range effort after collecting a Dusan Tadic cross just before the break.
The Serbian, signed as Adam Lallana's replacement in 2014, saw a threatening cross-shot cleared soon after play resumed, with Southampton pressing well and frustrating the Reds.
Daniel Sturridge's claim that he was upended in the box was ignored by referee Neil Swarbrick and Firmino saw a long-range effort stopped with ease, before Klopp turned to star man Philippe Coutinho.
Out since November 26 with an ankle injury, the Brazilian returned as a 61st-minute substitute in a bid to inspire the visiting side.
Coutinho was soon thrilling the raucous travelling support with his skill in winning a corner, but Saints broke at speed when that set-piece came to nothing and looked set to score, only for right-back Cedric to try his luck rather than cross.
In the longer run, the introduction of Coutinho was unable to inject much sustained attacking impetus to struggling Liverpool.
Saints continued to look the bigger danger and, after Rodriguez had an effort, substitute Shane Long beat Lucas Leiva on the left. Redmond met his low cross with a chipped effort that beat Karius but not the crossbar, with Lovren clearing on its way down.
Sturridge and Coutinho blazed over at the other end in a desperate attempt to level, but they were unable to cancel out Southampton's advantage.
Liverpool manager Jurgen Klopp was criticised for his selection when the youngest team in the club's history were held to a goalless draw by Plymouth at the weekend. Even with Manchester United up next on Sunday he will not employ the same approach, with most of his big names returning, although he may still retain a couple of youngsters.
HOW TO HANDLE COUTINHO?
Liverpool's influential playmaker Philippe Coutinho is in desperate need of pitch time after almost seven weeks out with ankle ligament damage if he is to stand a chance of starting at Old Trafford at the weekend. Klopp is keen not to overload the Brazilian too soon, so expect a 20-minute substitute appearance at best.
IS LORIS KARIUS FOR KEEPS?
The summer signing gets another chance to stake his claim to be first choice after a totally inactive game against Plymouth. The German was dropped a month ago after a couple of errors led to growing scrutiny and still has some convincing to do, but a run to a cup final would provide a much-needed confidence boost.
HAS CLAUDE PUEL GOT TIME TO FIGHT BATTLES INSIDE AND OUT?
Saints have not won since mid-December, so their manager, with issues to address on the pitch, could have done without captain Jose Fonte handing in a transfer request. The centre-back has been dropped for this match, but Puel will want to avoid a drawn-out row with the defender at such a pivotal time of the season. Continued speculation over the future of Fonte's defensive partner Virgil van Dijk is also not helping.
REVENGE IN THE AIR FOR SAINTS?
A large number of Southampton's current side were present for their 6-1 mauling at the hands of Liverpool in the quarter-finals of this competition last season, when they were torn apart by Divock Origi and Daniel Sturridge. The bad news for them is that Liverpool are playing better this year than last and Southampton's best chance of progress may well be to keep things tight to ensure they are still in the tie when they go back to Anfield later this month.
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