The Dutch star nodded home his fourth goal in four matches on 54 minutes to earn three vital points for Rafael Benitez's men after Sotirios Kyrgiakos had been shown a straight red card for a first-half challenge on Marouane Fellaini.
The Toffees had rarely looked like snatching a draw and their misery was compounded in injury-time when Steven Pienaar received a second yellow card.
The win clinched the Reds' fourth Barclays Premier League double over their neighbours and means they are now up to fourth in the table - albeit having played more games than their rivals.
With Aston Villa and Tottenham going head-to-head in the evening kick-off, Benitez and co went into the 213th Merseyside derby knowing that victory would not only cement their place as the number one club in Liverpool for another season, but also enhance their bid for the fourth Champions League spot.
The added spice meant that there was more than a hint of passion around the ground as the teams made their way into gladiatorial combat, and it was no surprise that Jamie Carragher won the game's first 50-50, crunching Pienaar with a well-timed tackle that the Kop greeted with a roar of approval.
It set the tone for a full-blooded clash that was rife with the sort of challenges that make these encounters an occasion to savour.
Everton came into the contest on the back of an impressive run and looked the more threatening early on, with a number of set piece deliveries causing problems for Pepe Reina, who was forced to contend with the dazzling glint of the mid-afternoon sun.
However, it was Tim Howard who was the first goalkeeper called into action, fisting away Steven Gerrard's driven free-kick from the far left of the penalty area.
It seemed to lift the hosts and another thunderous challenge from Carragher led to a corner that eventually saw the returning Daniel Agger crash a 12 yard volley over the top.
Things were warming up nicely and the Reds were beginning to enjoy the better of it. A lightning break almost saw Kuyt play David Ngog through before Emiliano Insua, who was making his 50th appearance for the club, rifled a shot just clear of the crossbar.
Everton responded and Leighton Baines tested Reina with a 25 yard free-kick before a two-footed tackle by Kyrgiakos saw the game swing the way of the Blues on 33 minutes.
The Greece international was playing in his first Merseyside derby, but it would not be one to remember, as referee Martin Atkinson chose to show him a straight red card and bring his participation to a premature end.
Any thoughts that Liverpool would be forced onto the back foot were appeased, however, as the home side ended the half strongly.
A good spell of possession saw Gerrard hauled to the ground and from the resulting free-kick the skipper curled a magnificent 20 yard free-kick against the bar, with Howard seemingly well beaten.
Tim Cahill then headed a glorious chance over the top deep into first-half injury-time to ensure the first 45 would end goalless.
Back in March 2006, Benitez's men had faced a similar situation to the one they were now in, after overcoming the early sending off of Gerrard to emerge 3-1 victors.
A repeat showing was the order of the day, and as Everton restarted the game attacking the Kop end, it was clear that much would rest on the young shoulders of Ngog.
The Frenchman was ploughing a lone furrow up top, but almost snatched the opener with a solo effort on 53 minutes when he raced onto Maxi Rodriguez's lobbed pass, held off Sylvain Distin and smashed a deflected shot just wide.
The resulting corner did not lead to the opening Liverpool craved, but two minutes later Gerrard won another one and sent in a delicious centre that Kuyt was on hand to flick home from just inside the six yard box.
It was the Dutchman's 10th of the campaign and saw him become only the third Liverpool player to score in both Merseyside derbies in the same Barclays Premier League season.
A goal behind, but a man to the good, the anticipated Everton onslaught was not evident at first and Ngog went close to making it two when he came within inches of converting Gerrard's centre at the far post. It was the striker's last contribution of the match before being replaced by Ryan Babel.
The Blues had an array of attacking flair in their line-up, but looked devoid of ideas as they searched for a way back into the game.
Indeed, it was Benitez's side that looked the more likely and Reina was only called into action in the dying seconds, when he was forced to deny a 30 yard drive from substitute Yakubu.
David Moyes' men threw caution to the wind in stoppage time, but after Pienaar saw red, Liverpool easily withstood the five minutes to stretch the Toffees' Anfield record to just two wins in their last 24 visits.