Liverpool broke Luton's hearts in an FA Cup classic at Kenilworth Road.
The visitors went in front when Steven Gerrard curled in but Luton struck twice as Steve Howard slotted home and Steve Robinson brilliantly drove in.
Djibril Cisse missed a penalty, Kevin Nicholls' spot-kick made it 3-1 before Florent Sinama Pongolle drove home.
Xabi Alonso lobbed in from 40 yards to level, Pongolle headed past Beresford and at the end Alonso amazingly curled into an empty net from fully 60 yards.
It is a game that will surely go down as one of the most remarkable in the magical history of the world's oldest cup competition.
Liverpool brought in keeper Carson for Jose Reina and his handling was tested early on as Luton started at a frantic pace.
But the first time Gerrard got time and space on the ball, he punished the Championship side.
He found room on the right-hand edge of the Luton box and when Cisse gave him the ball, Gerrard wonderfully arched it away from Marlon Beresford's despairing dive and into the corner.
To their credit Luton refused to buckle after the goal and Howard should have levelled, but he slipped and scuffed wide after the impressive Vine skinned Carragher down the left.
They got a deserved equaliser when Robinson sent Howard through and he kept his composure, coolly rounding Carson and slotting home from 10 yards.
Markus Heikkinen's header was cleared off the line by Steve Finnan as Luton began to dominate and, to Liverpool's horror, two minutes before the break Mike Newell's side were ahead and in dreamland.
Carlos Edwards fed Robinson and his sensational first touch took Carragher out of the game, the little striker rifling past Carson to send Kenilworth Road into raptures.
Luton were on cloud nine at the break, but they nearly came crashing back down to earth immediately after the interval.
Referee Mark Clattenburg gave Liverpool a penalty for Paul Underwood's foul on Gerrard, but Beresford dived to his right to save Cisse's 12-yard effort.
Three minutes later, Nicholls showed Cisse how it should be done, stroking home after Carson had adjudged to have brought down Vine in the box.
Liverpool brought on Pongolle and he made an immediate impact, reducing the deficit with a well-taken low drive.
Luton were trying to hold out as the Reds surged forward and they levelled when Alonso shot from 40 yards, his looping, swirling effort sailing into the roof of the net past a bemused Beresford.
The Hatters' fell behind when Finnan crossed from the right and Pongolle nodded past Beresford.
Their misery was compounded when Beresford went up for a corner, Liverpool cleared and Alonso brilliantly drove the ball into the Luton net from well inside his own half.
Luton: Beresford, Foley, Coyne, Heikkinen (Barnett 74), Underwood, Edwards (Feeney 81), Nicholls, Robinson, Brkovic (Showunmi 81), Howard, Vine.
Subs Not Used: Morgan, Brill.
Goals: Howard 31, Robinson 43, Nicholls 53 pen.
Liverpool: Carson, Finnan, Hyypia, Carragher, Riise, Kewell, Gerrard, Sissoko (Sinama Pongolle 57), Alonso, Crouch (Kromkamp 79), Cisse (Warnock 89).
Subs Not Used: Hamann, Reina.
Booked: Sissoko, Carson.
Goals: Gerrard 16, Sinama Pongolle 62, Alonso 69, Sinama Pongolle 74, Alonso 90.
Ref: M Clattenburg (Co Durham).