Blues defender Yobo turned Javier Mascherano's speculative shot past Tim Howard and into his own net after just 12 minutes of a tense contest.
 
Everton came back strongly, with Pepe Reina forced to produce a world class double save to thwart David Moyes' side midway through the second half.
 
It was a crucial moment as Kuyt swept home late on after Howard had failed to hold Albert Riera's effort to ensure it's the Red half of the city who will enjoy the local bragging rights, as well as move Liverpool up to fifth in the Barclays Premier League table.
 
After the midweek victory over Debrecen in Budapest, Rafa Benitez opted to name an unchanged line-up as Liverpool went in search of a third consecutive win over Everton at Goodison Park for only the third time in the club's history.
 
With Fernando Torres not included in the Reds' squad, David Ngog was handed a derby debut in attack, supported by Steven Gerrard.
 
There were also derby bows for Glen Johnson and Emiliano Insua in the Liverpool starting XI.
 
The blood and thunder approach to these games is what sets the fixture apart from the numerous other derby clashes around the country - and this one was no different with both teams clearly fired up for the occasion right from the first whistle.
 
As too were the two sets of fans, both equally making an ample din inside the stadium, with each and every incident and challenge greeted by a torrent of noise raining down from the terraces.
 
Both Reds and Blues had early half chances, with Ngog and Diniyar Bilyaletdinov failing to control when in promising positions.
 
But it was Liverpool who took the lead after 12 minutes. A sterling interchange by Fabio Aurelio and Insua down the left created a shooting opportunity for Javier Mascherano 20 yards from goal.
 
The Argentina captain needed no second invitation to dispatch an effort towards Tim Howard's goal, which appeared to be heading wide, but the ball took a wicked deflection off Blues' defender Joseph Yobo before nestling into the bottom corner.
 
Mascherano set off on a wild charge towards the pocket of delirious Liverpool fans gathered in the Lower Bullens, though the strike will go down as an own goal by Yobo.
 
Moments later, Gerrard burst beyond Sylvain Distin and drilled an effort into the palms of Howard from the right corner of the box.
 
However, Everton should have levelled on 17 minutes when the unmarked Bilyaletdinov contrived to skew wide from point blank range after Marouane Fellaini had flicked Tony Hibbert's long throw into the box to the back post.
 
It was a real let-off for the visitors and the Russian winger brought a diving save out of Reina soon after with a snap-shot from another deep centre before Steven Pienaar lashed an effort over the bar as Everton pressed forward in search of a quick riposte.
 
David Moyes' side were causing problems in the air and Jo could not get enough purchase on another Fellaini nod on, which enabled Reina to gather with ease.
 
Everton thought they had finally beaten the Anfield custodian on 33 minutes when Jo fired Bilyaletdinov's clever reserve-pass into the back of the net from 10 yards, but the striker was in an offside position and the effort was chalked off.
 
Liverpool had offered little in terms of an attacking threat after taking the lead, but only a fantastic save from Howard prevented the visitors from extending their advantage four minutes before the interval as the American stopper punched Insua's close-range header from Johnson's cross to safety.
 
Back came the Blues and Jo again thought he had levelled, prodding in from a corner, but the Brazilian was once more thwarted by the raised flag of the referee's assistant to ensure the Reds went into the break in a winning position.
 
The second half began in similar fashion, with Everton enjoying the lion's share of possession. Pienaar almost left Reina red faced with a stinging drive that the goalkeeper spilled between his legs, but managed to reclaim the ball before it crept over the line.
 
Agger glanced a Gerrard corner over the bar as Liverpool threatened for the first time after the break, before brilliant work by Dirk Kuyt to win back possession created a chance for Ngog inside the six-yard box, but a flying block by Yobo prevented what appeared to be a certain goal.
 
But Reina kept Liverpool's lead intact with a stunning double save 20 minutes from time to first deny Tim Cahill's header from John Heitinga's centre before repelling Fellaini's follow-up at close-quarters.
 
It was simply world class from Liverpool's goalkeeper and it proved decisive as Kuyt added a second for Benitez's charges ten minutes from time.
 
Gerrard did brilliantly to direct possession away from Hibbert into the path of substitute Riera, who drilled a low effort towards goal.
 
Howard was able to save, but he couldn't hold onto the ball, allowing Kuyt to sweep home from yards out.
 
Only a good stop from the Everton No.24 prevented Liverpool from furthering their lead via Riera, but the second goal had broken the Blues' spirit and the Reds were able to claim a morale-boosting victory and extend their unbeaten run to five matches.