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It was the Toffees first derby win for five years and takes them to the lofty position of second in the Premiership and two points clear of champions Arsenal.
Many of the critics who predicted they would fall flat on their faces are now being forced to take them seriously as title contenders.
Before the game Moyes played down the importance of victory saying: "Derbies are unpredictable. There is a great deal of excitement and always a lot of twists."
"We're just happy to have 33 points from 16 games and I'd rather have that than one derby win."
But no doubt secretly his heart was yearning to break his duck since taking the reigns at Everton and now he can truly revel in the adulation of his fans.
The 200th Mersey derby got off to a tame start with both sides feeling each other out while the Goodison faithful waged an energized chant battle with the Liverpool fans who had travelled the short walk across Stanley Park.
But after a few early over-zealous tackles the two fierce rivals quickly caught their fans enthusiasm for the match.
It was Liverpool who created the game's first real chance.
Australian international Harry Kewell, who scored his first club goal against Everton, picked up the ball just outside the area and made a decisive charge towards goal.
He tricked Alessandro Pistone with a clever turn before unleashing a left-footed shot which was easily handled by Nigel Martyn.
But Everton were quick to respond with a well crafted move which should have been converted by Tim Cahill.
Thomas Gravesen made a tough run through three red shirts before laying the ball off to Marcus Bent who was sprinting down the right wing. He ran to the baseline before lofting a perfect cross to the unmarked Cahill in the centre of the box.
But Kewell's fellow countryman's header drifted well wide drawing an agonizing groan from the crowd.
The visitors nearly took the lead with a series of scrappy shots in the 34th minute
Champions League goal hero Neil Mellor saw his header cleared off the line
The ball then fell to Steven Gerrard whose effort was again stopped and his effort was finally headed over the bar by Mellor.
The play perfectly summed up the nature of a lacklustre first half and both sides trudged off the pitch for their team talks.
Gerrard; Liverpool's captain fantastic and scorer of the winning goal against Olympiakos midweek suffered a nightmare of a game.
His usual pinpoint passing was constantly way off the mark and most of his shots fell wide.
One particularly cringeworthy moment came when he stepped up to take a free-kick just yards outside Everton's area. The blues must have been holding their breath but the England midfielder slipped, falling on his backside and skied the ball into the stands, much to the delight of Toffees fans.
And as is often the case when Gerrard underperforms, so did the rest of his side.
Antonio Nunez replaced Dietmar Hamann in the 66th minute and his first touch should have softly carried him towards threatening Everton's goal, but again instead it sailed out of bounds.
The Blues took advantage of Liverpool's dip in confidence and temperament.
Lee Carsley crossed the ball from the left wing towards Bent who quickly switched it to Leon Osman.
But Carsley had made a sneaking swooping run through the middle and Osman, using great peripheral vision, found him on the edge of the box.
He placed a brilliant, curling strike past Chris Kirkland to break the stalemate and carry his season's goal tally to four.
A late surge of good play from Liverpool in the last 15 minutes was too little too late.
A Gerrard run into the box ended with up a rare shot on target which produced perhaps Martyn's best save of the match.
An injury-time pitch invader momentarily delayed Liverpool's misery as the stewards rugby tackled him to the ground and carted him off.
But the final whistle eventually sounded and Everton had clearly not only won their first derby at home in seven years, but also burst Liverpool's Champion's League confidence bubble.
After the game Everton's goal scorer Carsley described the atmosphere in the changing room as like winning the FA Cup.
He said: "Everyone's been waiting for us to fall over but we keep going and going.
"We've got great team spirit and we feel we can do damage to any team.
"It was an important match for us and the fans. It feels like we've just won the FA Cup."
Liverpool manager Rafa Benitez said his side was unlucky to concede the goal but also praised Everton's tough performance.
He said: "We had two or three clear chances and we failed to score it is as simple as that.
"This was always going to be a difficult game and we paid the price for a lapse in concentration. For me we controlled the first half and we were unlucky to concede a goal.
"If we had scored in the first half it might have been a different story.
"But Everton are a good team and I think people have to consider them as genuine title contenders now."
Moyes praised his team's performance and said this win was a great example of how far his team had evolved.
He said: "There have been a lot of satisfying games recently but a derby win is that little bit extra special.
"An Everton team haven't beaten Liverpool at home for seven years and I believe five years overall.
"This is a good pointer for how far we've come, last year we were at the bottom of the table but now we're at the top.
"This is an exciting position for us to be in at the moment but we must remain humble because that can change.
"All we can do is to promise the fans that we will try hard to keep winning and we will."
Man of the Match: Thomas Gravesen - The Danish powerhouse gave another outstanding display in the Toffees engine room and did much to nullify the threat of Reds skipper Steven Gerrard.