Liverpool manager Rafael Benitez hailed his side's 1-0 victory in the 213th Merseyside derby as "massive".
Dirk Kuyt's close-range header 10 minutes after half-time proved the difference as the Reds played the last 55 minutes with 10 men after Sotirios Kyrgiakos was dismissed for a two-footed challenge on Marouane Fellaini. Everton failed to capitalise on their numerical advantage and they had Steven Pienaar sent off for a second bookable offence in the fifth minute of second-half stoppage time.
"It was massive for us. To win in a derby is really important - to win in a derby with 10 players at Anfield is more important," said Benitez.
"I was really pleased for the players and the fans because everyone was working so hard that I think at the end we deserved to win.
"From the beginning we were doing well in terms of work-rate and after (the sending-off) it was amazing to see all the players working so hard, defending and also going forward and trying to score."
Benitez said he did not want to comment on either the performance of referee Martin Atkinson or the ferocity of some of the challenges which went in on his players.
"I think there were too many tackles which you can analyse so I prefer not to speak about the referee," he added.
"You can talk about 20 different tackles and you will have different opinions so I prefer not to say anything.
"We were trying to win, they were trying to win - it is always the same in the derby.
"We knew it could be tough but we tried to do our best, we tried to control things and play with our heads not just with our hearts but sometimes it is difficult."