It was the Reds' eighth win in their last 10 league trips across Stanley Park and revenge for their FA Cup exit when the clubs last met back in February.
However, the scoreline was undoubtedly harsh on Everton who bossed the game for long periods and missed a string of chances.
Only some woeful finishing and expert goalkeeping from Pepe Reina kept the hosts at bay before Kuyt's goal 10 minutes from time settled it.
Victory re-ignites the Reds' push for a top four finish and leaves the Blues dangerously close to the Premier League drop zone after a run of just one win in their last 11 games.
Both clubs went into the game having taken just five out of the last 18 points on offer and in desperate need of a boost.
Rafa Benitez kept faith with the side which beat Debrecen 1-0 in midweek, while David Moyes made four changes following the Blues' 3-2 defeat at Hull.
Toby Hibbert was preferred at right-back to Lucas Neill and Jo was recalled up front as the Everton boss reverted back to playing 4-5-1.
The home midfield was boosted by the return from suspension of Marouane Fellaini and Diniyar Bilyaletdinov.
Yakubu and Louis Saha dropped to the bench, while Jack Rodwell was ruled out with a groin injury.
In a typically frenetic opening chances came and went at both ends with David Ngog shooting tamely at Tim Howard, while Bilyaletdinov failed to convert at the back post after Tim Cahill flicked on Hibbert's cross.
However, it was Liverpool who made the breakthrough in the 12th minute.
Javier Mascherano let fly from 25 yards out and his shot took a wicked deflection off Joseph Yobo and left Howard helpless as it flew into the bottom corner.
The Argentinean midfielder couldn't contain his delight and raced to the other end of the ground to celebrate in front of the visiting fans.
Everton refused to let the goal deflate them and they responded positively to that setback.