Rafa Benitez believes the 1-0 derby triumph could have a 'massive' impact on the season after Liverpool climbed to fourth place.
Dirk Kuyt won the match with his 50th Liverpool goal following the first-half dismissal of Sotirios Kyrgiakos.
Asked how significant the result could prove, Benitez replied: "It's massive for us. To win in a derby is always important, and to win with 10 players at Anfield makes it even more important.
"The people are really, really pleased - and I'm pleased for the players and the fans because everyone was working so hard. We deserved to win.
"From the beginning we did well in terms of workrate and after the sending off it was amazing to see all the players working so hard defending and also trying to go forward and score goals.
"Hopefully we'll stay in fourth but it's a long race so we have to keep going, keep working hard and winning games. At least we have won this and we can enjoy the next two or three days."
Benitez paid tribute to Kuyt after his fourth Merseyside derby goal secured victory.
"He's scoring some goals against Everton," said the gaffer. "It's good for him and good for our fans that he can be a hero. Also, 50 goals is an amazing record for a player who plays everywhere."
Kyrgiakos's 34th minute dismissal came after a two-footed tackle on Marouane Fellaini, who might also have seen red for his part in the incident.
Steven Pienaar then received a second yellow deep into injury time.
Benitez would not be drawn on the performance of referee Martin Atkinson. Instead he lauded the workrate of his players for triumphing with only 10 men.
"There were too many tackles which you could analyse, so I prefer not to comment about the referee," said Benitez.
"We tried to play with our heads as well as our hearts but it's difficult. We won two years ago with 10 players. Sometimes the team spirit is even better when you know you have to fight and stick together."
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