Captain Steven Gerrard opened the scoring with a 21st-minute header before Daniel Sturridge scored twice in quick succession - also missing a penalty in the second half - and Luis Suarez added the fourth five minutes after the interval.
Anfield's biggest derby win since 1972 opened up a four-point gap between the fourth-placed Reds and their near neighbours in sixth.
"I think all wins are important but especially against your rivals and psychologically it was a big win and can re-iterate the confidence and belief in the squad," said Rodgers.
"We have to keep our foot to the floor and keep these performances and keep scoring goals.
"We have a real focus and concentration going into every game and this was a massive game so for us to score four goals and defend like we did and attack with that efficiency was very pleasing.
"I think we have shown all season we are going to be in contention for fourth place."
Aside from missing the penalty Sturridge also incurred the anger of Suarez after opting to shoot instead of failing to pass to his better-placed strike partner and soon after was substituted.
The England international appeared to show a degree of petulance with his manager but Rodgers played that down.
"It wasn't a difficult decision to take him off. I thought his two finishes were sublime, he was outstanding in the game and he is always a threat," he said.
"I just wanted to shore the game up. I think he was so frustrated after the penalty. He has apologised to the group.
"I've told the players we have to show respect for the club and if a manager decides to take you off it is not always because you are playing poorly.
"When you come off the field you sit down and if you have a problem and want an honest answer come and knock on my door the next day and I'll tell you.
"He was frustrated he missed a penalty and was desperate to get on the scoresheet again but for me it was a brilliant individual performance."
Rodgers also dismissed suggestions he was unhappy with the club's hierarchy over transfer policy and said he still hoped to make a major signing before the window closed.
"I think we will hopefully do something over the next couple of days," he said.
"I saw some headlines after yesterday's press conference in terms of the manager being angry with the club - nothing is further from the truth.
"We are not just going to get the players we want to get in for a number of reasons.
"Hopefully we can have a major signing before the window shuts and if we can do that we will be set up for the remaining games."
Everton boss Roberto Martinez felt fielding the likes of Ross Barkley, Steven Pienaar and Antolin Alcaraz who were short of match fitness played into Liverpool's hands.
Added to their woes was the injury to striker Romelu Lukaku who injured ankle ligaments when Gareth Barry collided with him.
"He has his ankle in a cast. We will assess it but it doesn't look too good at the moment," said the Spaniard of the injury.
"We conceded the opening goal from a dead-ball situation from which Romelu picks up what appears a nasty injury and it seemed we put ourselves in Liverpool's hands and we lost concentration and our normal solid outlook.
"We got ourselves in a position where we lost the game in naive way.
"We were desperate to come here in the derby and be ourselves and give a performance our fans could be proud of but we got ourselves in a position where we played with our heart at times.
"It was nowhere near good enough for our standards."