They followed up the four they put past Toulouse in the Champions League on Tuesday with two goals apiece from Spaniards Fernando Torres and Xabi Alonso, Ryan Babel's first for the club and one from substitute Andriy Voronin.
It is the first time Liverpool have led the top flight for five years and their best start to a league season since 1998.
• Rafa: Babel goal was pick of the lot
You had to feel sorry for Derby. Fresh up from the Championship, they simply could not cope with the variety, pace and skill Liverpool possessed - or indeed the pots of money that has seen boss Rafael Benitez build his squad.
Benitez made five changes from the side that beat Toulouse in midweek, with Peter Crouch not even on the bench and Dirk Kuyt and Torres being paired up front.
Derby duo Claude Davis and Tyrone Mears both passed fitness tests for the Premier League's bottom club, but Dean Leacock missed out with a toe problem.
Steven Gerrard and Jamie Carragher watched from the directors' box, as both were still injured for Liverpool, who recalled Steve Finnan, Alonso, Jermaine Pennant and Babel along with Torres.
Without Paco Ayestaran - his assistant for 11 years at three clubs - alongside him, Benitez watched his team start with confidence and cruise through the game.
Ayestaran has shocked the Liverpool backroom by deciding he wants to leave, after what is believed to be several recent differences with Benitez.
How this affects the smooth running of Liverpool long term remains to be seen, but the players set about finding the gaps in Derby's rearguard with single-minded determination.
Daniel Agger saw a 25-yard shot clip Bob Malcolm and force Stephen Bywater into a flying save. The goalkeeper was in action again when he fielded Torres' header powered in from Pennant's cross.
Mears ventured forward to blast a 20-yard shot over the bar, and Stephen Pearson wasted possession after robbing Javier Mascherano and chasing 40 yards, before a poor pass let Liverpool off the hook.
But that was about it from Derby in the first period as they became increasingly occupied by the Red tide coming their way.
Kuyt cracked a 25-yard shot just over the angle and Mascherano saw a long-range effort skim the far post.
Derby were battling hard, trying to restrict the space in front of their defence, but it was becoming increasingly difficult and Mears was warned by referee Alan Wiley for a succession of fouls.
After one on Pennant, Alonso took the free-kick 35 yards out on the left. As his low drive swerved into the box, Kuyt ran across Bywater's line of vision and the ball nestled in the back of the net.
Kuyt should have made it two when Pennant raced away on the left to lift in a cross for the unmarked Dutchman to head over from eight yards.
Two minutes later Pennant's pace and skill severely embarrassed Malcolm and, from yet another cross, Kuyt was there again to lift his effort over the bar.
Liverpool were pouring forward now, and after one sustained run from Finnan, the cross was controlled superbly by Torres before the striker cracked his effort over the top.
Derby were praying now to get to the break without further damage, but they were to be disappointed.
The second goal came in first-half injury-time when Pennant set up Babel, who side-stepped two diving defenders before belting in his first goal for the club.
The second period started just as ominously for Derby. Pennant, Torres and Kuyt combined to set up Alonso from 10 yards, but the Spaniard lifted his effort into the Kop.
The third was not long coming. Mascherano dispossessed a dithering Malcolm 30 yards out and pushed the ball to Torres. The Spaniard then took on and beat three defenders in a run across the box before burying his shot into the bottom corner on 56 minutes.
Three minutes later the hapless Malcolm was withdrawn to save him further punishment and Gary Teale came on.
Liverpool were playing at such a pace and with outstanding control, Derby just could not handle it and continually gave the ball away or made forced errors.
Liverpool's response was to take off the excellent Pennant, to a rousing ovation, with Yossi Benayoun coming on.
Derby then took off Mohammed Camara - another defender out of his depth - and replaced him with James McEveley.
Andy Griffin was booked for pulling back Torres on 65 minutes, with the Liverpool attacks incessant.
The fourth goal came on 69 minutes when Babel crossed from the right, Benayoun's effort was blocked and Alonso placed an 18-yard shot into the bottom corner.
Next on was Voronin in place of Babel and virtually his first touch produced the fifth. Kuyt surged into the box on the right and fired in an angled shot that Bywater could only push out for the Ukrainian to force home from a couple of yards.
It got worse for Derby a minute later, when a poor back-pass by Andy Todd was snapped up by Torres, who rounded Bywater to run the ball into an empty net.
Mohamed Sissoko came on for Mascherano between the fifth and sixth goals, Darren Moore being sent on by Derby for Griffin on 80 minutes, and Derby survived without any further damage.
Rafa: Babel goal was pick of the loLiverpool boss Rafael Benitez hailed his side and the quality of their performance after hammering hapless Derby 6-0.It was their biggest winning margin at Anfield since 1999 and saw them take over at the top of the Premier League for the first time in five years.
They could well lose that lead on Sunday when Chelsea play at Aston Villa, but Benitez knows that his side's unbeaten start to the season is creating quite a stir.
The Reds have managed three victories in a week - all without skipper Steven Gerrard - and scored 12 goals into the bargain against Sunderland, Toulouse and Derby, without conceding.
Benitez said: 'In the first half we played really well, we found space, passed well and had some excellent movement.
'I was pleased with all the goals, but Ryan Babel's was probably the best. But it was good to see Fernando Torres keep scoring, while Xabi Alonso could have had a hat-trick.
'When (Andriy) Voronin came on he fitted straight into the play and also scored. I just wanted to see Dirk Kuyt score because he had done so much running and created a lot of space for other players.
'It was good to score four and then six in front of our own fans this week.'
Benitez's only cloud on the week has been the shock departure of his assistant Paco Ayestaran, who quit after what has been a gradual breakdown in the relationship between the two men, who have worked together for 11 years with Tenerife, Valencia and Liverpool.
Liverpool are currently working on the details of Ayestaran's departure, having signed a new contract only last season.
Benitez added: 'Personally this has been disappointing. But we are determined to keep going, nobody is bigger than the club.
'It is a shame this has happened but the club are now talking to Paco over the terms of his departure.'
Derby boss Billy Davies refused to be too downhearted after such a drubbing.
He said: 'I am very confident that in time we will get it right in this division.
'I have said that our season will not really start until the transfer window ends, now it is over and we have got players in.
'We had six players missing through injury, I am a realist, I know the position we are in when we play a team like Liverpool.
'The result disappointed me. To show such a lack of discipline and care is very disappointing. We made silly mistakes and some of my players are not brave enough.
'That cost us dearly, one or two of them are freezing at this level, at this level they are showing themselves not to be brave enough.
'They are standing and admiring top-class players and letting them play.'
He added: 'All we can do is our best in games against teams like Liverpool. I will not criticise any one player, we are in this together.
'But it is good we are not playing for two weeks. We can get players fit and get new players who have arrived bedded in. Kenny Miller has arrived, he has a hamstring injury and now we can get him fit.
'He is a confident lad, a bit arrogant and he will be a threat up front for us.'
Liverpool 6 - 0 Derby County
