Southampton boosted their prospects of avoiding the drop by recording their first Premiership win under Harry Redknapp to frustrate Liverpool.
David Prutton took advantage of poor Liverpool defending to fire Saints ahead from Peter Crouch's return pass.
Prutton then turned provider, crossing for the unmarked Crouch to double their lead before the break.
Steven Gerrard shaved the bar for Liverpool in the second half but Saints defended stoutly for a vital win.
Southampton's cause was not helped by a training ground injury to Nigel Quashie, which ruled the £2.5m signing out.
A full-strength Liverpool team showed eight changes from the side that exited the FA Cup at Burnley.
But Liverpool appeared to still be suffering from that midweek hangover as Southampton took a fifth-minute lead.
Mauricio Pellegrino was caught in possession in his own penalty area by Prutton, who took a return pass from Crouch before drilling his shot under Dudek.
A foul by Calum Davenport on Milan Baros gave Liverpool a chance to chance to level quickly but they made a horrible hash of the free-kick as they tried to work a training ground routine.
Prutton confidently took advantage of the space as Liverpool backed off to fire a shot wide from 23 yards.
Prutton opened the scoring and laid on Southampton's second
Liverpool worked their first opportunity on 24 minutes when Baros picked out Gerrard but the England man blazed over from 18 yards.
It proved a costly miss as Southampton immediately doubled their lead. Anders Svensson, Jamie Redknapp and Mikael Nilsson neatly worked the opening to release Prutton whose cross found Crouch completely unmarked at the far post to head home his sixth goal of the season.
Liverpool were distinctly shaky at the back and a posse of four yellow shirts failed to stop Crouch winning Rory Delap's long throw, but the ball went through to Dudek.
Southampton should have put the game beyond reach on 33 minutes when Delap broke clear from midfield but with only Dudek to beat and with support to his right his tame effort gave the Polish keeper an easy save.
Dudek got his angles right to deny Crouch but at the other end Nilsson was caught in possession and Gerrard fed Luis Garcia but Antti Niemi was out quickly to block.
Florent Sinama Pongolle replaced Stephen Warnock at the start of the second half as Liverpool boss Rafa Benitez looked to fire his lacklustre side up.
Claus Lundekvam kept a tight rein on Fernando Morientes
Claus Lundekvam's block denied Morientes before Baros worked himself a yard of space to fire a shot inches wide.
Redknapp was harshly pulled up for handball on the edge of his penalty box and the resulting free-kick was touched to Gerrard whose shot skimmed the top of the bar.
It was a much-improved second half from Liverpool and after Lundekvam's challenge whipped the ball off Baro's toes, Morientes found space but blasted wastefully over.
But Southampton defended doggedly, in contrast to Liverpool whose panic-stricken defending almost presented Crouch with a chance as Pellegrino's clearance struck Carragher.
Referee Alan Wiley waved away strong Southampton penalty appeals when subsitute David Raven appeared to shove Crouch as a cross came in.
Baros was Liverpool's main threat and he checked back on to his right foot to fizz a shot just wide of Niemi's right-hand post and as Southampton defended deeper, Gerrard drove forward but fired wide from 20 yards.
Southampton: Niemi, Telfer, Lundekvam, Davenport, Higginbotham, Nilsson (Jakobsson 76), Prutton, Delap, Redknapp, Anders Svensson (Jones 89), Crouch.
Subs Not Used: Oakley, Smith, Ormerod.
Goals: Prutton 5, Crouch 22.
Liverpool: Dudek, Carragher, Hyypia (Raven 67), Pellegrino, Warnock (Sinama Pongolle 45), Luis Garcia, Gerrard, Hamann (Biscan 78), Riise, Baros, Morientes.
Subs Not Used: Traore, Carson.
Booked: Hamann, Luis Garcia.
Ref: A Wiley (Staffordshire).