Steven Gerrard gave the Reds the lead after quarter of an hour when he received the ball from Cisse on the right and put a brilliant shot into the top left hand corner of the net.
The goal was the only moment of quality from Liverpool in the first half however. The Reds looked tired and off the pace for the rest of the half and Luton troubled Rafa Benitez' side more than any other side has done this season.
Luton got the rewards their efforts deserved when Steve Howard equalised after half an hour. The Luton striker picked up a pass, rounded Scott Carson in the Liverpool goal and slotted home the equaliser.
It got worse for Liverpool just before half time when Steve Robinson brilliantly turned Jamie Carragher and fired in Luton's second from just inside the 18 yard box.
Liverpool looked a better side when they emerged for the second half and Harry Kewell, the only Liverpool player who looked on the pace in the first half, denied an equaliser by a fingertip save from Marlon Beresford.
Liverpool were then awarded a penalty when Steven Gerrard was brought down in the box after running onto a Harry Kewell pass. Djibril Cisse stepped up to take the spot-kick but saw it well saved by the Luton keeper.
Things got worse for Liverpool when Luton immediately went up the other end and were awarded a penalty of their own when Carson brought down Vine although tv replays later showed that the challenge was outside the box and it was Vine that had stamped on Carson before losing his balance and going over.
Nicholl scored the penalty to make it 3-1 and suddenly Liverpool were facing a second successive third round exit to Championship opponents. However, as they showed in Istanbul in the European Cup final, the Reds are never beaten and the fightback began.
Florent Sinama-Pongolle was introduced as a 57th minute subsitute and he made an immediate impact, latching onto a Gerrard through ball in the 62nd minute and drilling a low shot into the bottom left hand corner.
Xabi Alonso scored the equaliser in that fantastic game at the Ataturk Stadium in Istanbul and he did the same again here. A 35 yard volley from the Basque flew in over Beresford's head and the Reds were on terms after 69 minutes.
It was 4-3 to Liverpool five minutes later when Pongolle scored his second of the game when he rose to meet a Riise cross and headed in to put the Reds back in the lead.
Luton had a corner in injury time and they pushed everybody forward to get an equalier, including goalkeeper Marlon Beresford. When Liverpool cleared the corner Xabi Alonso picked up the ball and was able to score in an empty net from inside his own half with the final kick of a pulsating cup tie.