The Scot has never won at Liverpool in 21 attempts, drawing just seven, with the Toffees, Manchester United, Sunderland and West Ham.
In his second spell at Everton he would love nothing more than to end that sequence, having enjoyed a nine-match unbeaten Premier League run since taking over in January.
Asked whether the unwanted record was a weight around his neck Moyes said: "Yeah. I don't want it, I want to win and I want to make sure I get rid of it if I can.
"Am I sick of it? I would be lying if I said I look forward to going to it all the time because it has been such a hard place to get results.
"It's nothing to do with the surrounds, nothing to do with the pitch, nothing to do with anything else, it is to do with them always producing good teams.
"I think every team in the Premier League, when they go to Anfield, has a game on their hands.
"We might be further away from Liverpool than we have ever been at the moment. By the time I had left here maybe 10 years ago we were much closer to Liverpool, we were competitive, competing around the same areas of the league at the time.
"I think at the moment it is the biggest gulf between the two clubs.
"It would be huge for us as a club to get that because it is not something the club have done particularly often."
Everton snatched a 2-2 draw at Goodison Park last month courtesy of James Tarkowski's goal deep into added time.
It was a game in which they managed to keep this season's Premier League top scorer Mohamed Salah quiet.
Moyes joked they could "try to build a wall or something" to stop the Egypt international, who is out of contract in the summer, but after acknowledging he was one of the Premier League greats the Everton boss said he would be glad to see the back of him.
"Yeah, we will enjoy taking him to the airport and getting him on the plane and getting him out (when he finally leaves Liverpool)," he said.
Moyes has had to deal with speculation about players of his own leaving with the future of centre-back Jarrad Branthwaite up for discussion after only making England's under-21 squad in the international break.
But the Everton boss denied the defender would have to leave to satisfy his ambitions.
"I would say to him: Phil Jagielka, Leighton Baines, Joleon Lescott. That's three who we brought in who went on to become international players for England," said Moyes.
"I think Jarrad is in that position where he could easily go on and do that. I see it as a challenge for Jarard to pick up and say, 'come on, I'm going to show what I can do'.
"But I think this is a club who develop quite a few England internationals to be honest over the years and we've got a very good goalkeeper (Jordan Pickford) who's playing for England as well."