Dynamic duo Tim Cahill and Mikel Arteta did the damage for the Toffees and the performance was not lost on Moyes who was revelling in the famous derby win.
"The smile's not come off my face. It's been a long time. I'm really disappointed I've not been able to win more derbies," he said.
"We play against a really good football club but it's not been because of a lack of effort.
"In the past we've probably lacked the quality to match Liverpool. I don't think we do now.
"We've got players of real quality who could play for the big clubs.
"Whoever scored the first goal was going to be really important and I think we always looked more likely in the first 20-25 minutes.
"I was delighted to get in front because both teams have not been scoring that freely."