An incredible game at Kenilworth Road saw Steven Gerrard give the Reds the lead in the 16th minute before Steve Howard and Steve Robinson efforts saw the Hatters go into the break 2-1 ahead.
Marlon Beresford saved a Djibril Cisse penalty before Kevin Nicholls made no mistake from the spot to put Town in the driving seat.
But two goals from substitute Florent Sinama Pongolle either side of a 35-yard rocket from Xabi Alonso left Luton broken-hearted.
The final stages saw Town goalkeeper Beresford in the Liverpool box to attack a corner, but as the ball broke clear it allowed Alonso to register his second long-range effort of the evening as he netted from the half-way line to end matters.
Hatters made just one change to the side that lost 2-1 to Watford in the opening game of 2006 as Chris Coyne returned from the dead leg he sustained in the New Year's Eve reverse at Ipswich to replace Leon Barnett at the centre of defence.
Rowan Vine kept his place in the side after shaking off a hamstring problem that saw him hobble out of the derby defeat.
Liverpool meanwhile made four changes to the side that drew 2-2 with Bolton in their last Premiership outing as they named a strong starting 11.
Town started brightly with Vine looking lively and forcing a corner off Mo Sissoko with a second minute cross.
It was Liverpool though that had the first attempt in the eighth minute when Sami Hyypia blasted wide from ten yards after Cisse had contested a Gerrard corner.
Town continued to press with Vine hitting a 15-yard volley at Scott Carson from Robinson's 12th minute cross on the left, but it was Liverpool who took the lead with a moment of brilliance from Gerrard four minutes later.
The move started with Nicholls losing out to Sissoko, who fed Cisse to cross for Crouch in the box. Liverpool's beanpole striker laid the ball back to Cisse, who teed up Gerrard to curl home magnificently with a first-time right-footed effort for his 16th goal of the season.
Harry Kewell almost found the opposite corner of the net four minutes later with a 20-yard effort that beat Beresford, but crept past the far post.
Luton should have equalised in the 25th minute. Vine did superbly well to beat Steve Finnan down the left and his cross found the unmarked Howard, who could only poke wide on the stretch from six yards.
Howard more than made amends in the 31st minute however when he set Kenilworth Road alight with a deserved leveller.
Ahmet Brkovic threaded the ball through and Howard beat the offside trap, rounded Carson and fired high into the net from 15 yards for his ninth goal of the campaign.
The goal inspired Luton to press further and they twice went close in the space of three minutes as Markus Heikkinen saw his header from Paul Underwood's 40th minute corner cleared off the line by Finnan before Vine saw a fierce 20-yard shot pushed away by Carson three minutes later.
Luton finally got the goal their pressure warranted when Robinson collected Edwards' short cross in the area, brilliantly turned Jamie Carragher and dispatched his second of the season past Carson from 12 yards.
The second period almost started disastrously for Town as Kewell saw his shot tipped over by Beresford 30 seconds in before racing clear four minutes later and squaring for the on-rushing Gerrard, who was bundled over in the box by Underwood for a penalty.
Cisse stepped up to take the kick, but Beresford dived to his left to superbly hold the striker's effort.
Instead Luton showed how a penalty should be taken as they stretched their lead in the 53rd minute. Howard robbed Hyypia to set Vine running free and the forward rounded Carson only to be pulled down by the goalkeeper.
Carson was lucky to stay on the field to face the kick, receiving only a yellow card, but he was powerless to stop Nicholls blasting home his fifth of the season.
Liverpool roared back with Cisse and Crouch going close before substitute Sinama Pongolle made it 3-2 in the 62nd minute as he ran on to Gerrard's pass and shot beneath Beresford from 20 yards.
The unrelenting action continued with Cisse driving into the side-netting from John Arne Riise's cross as Liverpool's superior fitness began to show.
The visitors then equalised with an extraordinary strike from Alonso in the 69th minute as the Spaniard's 35-yard thunderbolt caught out Beresford as it flew over the 36-year-old's head with the aid of a slight deflection and nestled in the back of the net.
A fantastic comeback was then complete in the 74th minute as Sinama Pongolle rose to head his second into the top corner from 15 yards after Gerrard's right-wing cross.
The Hatters pushed on until the end, but Alonso's second from the halfway line in the fourth minute of stoppage time when Beresford was up attacking a corner ended a spellbinding game of football.
Luton - Rowan Vine. The speedy forward often caught the Liverpool defence flat-footed and he was unlucky to be on the losing side in a pulsating game of football.
Liverpool - Steven Gerrard. Was once again an inspiring figure at the heart of the Reds midfield, continually driving the visitors forward even when the task seemed hopeless midway through the second half.