But in 16 games since then they have won just once and earned just seven points, leaving them hovering precariously above the bottom three.
Fernando Torres had a 10-yard angled sidefoot tipped over by keeper Boaz Myhill but having survived that early scare Hull quickly set about their task.
They snapped into tackles as they closed Liverpool down with an urgency and passion.
The in-form Yossi Benanyoun was one of the few Liverpool players able to keep possession and he went close when he curled a strike on to the roof of the net.
Hull responded with a Geovanni piledriver that flew past the angle of Jose Reina's goal and a Liverpool defence that has conceded eight goals in their last two matches was looking shaky.
Benitez unhappy with Hull's 'terrible' pitch
And Arbeloa had to clear a goalbound Dean Marney header as Liverpool continued to give Hull plenty of reason for optimism.
At the other end, Hull had a discipline and organisation that Liverpool were struggling to break through.
Alonso is renowned for his ability to pick a pass but was continually frustrated as he tried to pick a way through the Hull's defence.
In the end, it was not the Spaniard's guile and vision that breached the Tigers defence but pure power after the Reds were awarded a dubious free-kick for a George Boateng challenge on Javier Mascherano.
Alonso's initial effort was blocked by the wall but he half-volleyed the rebound unerringly into the bottom corner.
Hull's cause was not helped when Folan - being blocked from chasing a backpass by Skrtel - kicked out at the centre-back and was deservedly red-carded.
It got worse when Benayoun's cross was only cleared to Skrtel, whose shot was diverted past keeper Myhill by Kuyt's header.
Hull undone by 'poor decision' - Brown
Hull were clearly unhappy with the performance of referee Atkinson and felt there had been a handball by Lucas Leiva in the build up to Liverpool's second goal.
But the home side kept fighting and were rewarded with Geovanni's goal from Daniel Cousin's cross to set up a frantic finish.
Torres stuck the woodwork with a header before Hull's hopes were ended when Kuyt struck his second to briefly move Liverpool top on goal difference ahead of Manchester United's 5-2 victory over Tottenham.
Hull boss Phil Brown: "We were having to pick players up at half-time after an injustice was done.
"We knew it was going to be tough and somebody has let them off the hook. I cannot see a free-kick there.
"I have no excuses for the sending off. I feel it was justified and it will be dealt with in-house. That killed us, 10 against 11, and Caleb made it a little bit easier for them."
Liverpool manager Rafael Benitez hailed Dirk Kuyt: "Everybody is very pleased for him.
"Kuyt is a fantastic professional and is always working hard to score. He was very positive and we are really pleased for him.
"We were working very hard and we have to be satisfied. We were nervous because the pitch was terrible and difficult to pass on."
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