The Toffees have made a much better start to the season than city rivals Liverpool and go into the match six points ahead and looking to preserve an unbeaten record at Goodison Park this term.
Everton's record in derbies in the decade since Moyes arrived at the club has been one of his biggest disappointments, with the Scot having won only four of 23 meetings.
Last season the Toffees finished ahead of Liverpool in the table but lost all three matches against the Reds, including painfully at Wembley in the semi-finals of the FA Cup.
Moyes said: "I think we're much closer. I thought we went into the FA Cup semi-final maybe slight favourites because of our form. It didn't work out that way but I think over my time here we've been playing catch-up, which is right, we have to say that honestly.
"I think we possibly go into this game slightly ahead of Liverpool, but we need to show it, we need to prove it. We didn't do it in the game at Wembley so we have to try to do it in this one."
As well as form, Everton can also point to other potential advantages.
Unlike Liverpool, who played in the Europa League on Thursday, they have had a whole week to prepare, while Reds boss Brendan Rodgers will be experiencing his first Merseyside derby, which Moyes admits can take some getting used to.
He said: "The way the Premier League is, the games are so intense, I do think if you can have a free week then it's better, but if you asked me would I like to be in Europe, I'd rather be playing in Europe and going into a derby than not.
"Being in derbies of this magnitude is something that, once you do it, you actually realise what it's like. I was fortunate to play in Old Firm games which sort of set me up for what to expect in this game.
"I think being a manager in it is something slightly different but once you get your first one under your belt it makes you feel a lot easier. Brendan will have experience from Cardiff and Swansea derbies, which are fiercely contested."
Everton's success and Liverpool's struggles have certainly given an added twist to Sunday's match.
The relationship between the two has not had the same heat as the battles between Liverpool and Manchester United in recent years, with Moyes admitting his side have not been seen as enough of a threat.
The match will also be played out against the backdrop of the Hillsborough Independent Panel report, the outcome of which has been both cathartic and traumatic for the city.
Everton staged their own touching tribute to the 96 victims of the 1989 stadium disaster ahead of their game against Newcastle last month, which was very well received by the red half of the city.
Moyes said: "This is a city that has come together in the last two or three months with the news about the Hillsborough disaster. I think the city has pulled together but I still think the fans want the competitive spirit amongst them.
"They want to have a go at their pals if they're Liverpool supporters if we win and vice versa, so that hasn't changed. It's a great rivalry.
"Probably over the years Everton haven't done enough to compete against Liverpool and because of that maybe Liverpool haven't considered Everton that big a threat, and rightly so, but I think now we're getting a bit closer.
"The derbies are always good, we enjoy them, we've not had enough success in them, but we go into it in a good frame of mind, nice and positive, we've got a lot of self-belief."
Everton will be without suspended playmaker Steven Pienaar and injured midfielder Darron Gibson (thigh) but midfielder Marouane Fellaini (knee) and full-back Tony Hibbert (calf) could make welcome returns.