The Toffees endured a torrid start to the season and remain just four points above the relegation zone.
However, Moyes believes his squad posses the ability to turn their season around and is optimistic about the club's chances of finishing in the top half of the table, and possibly in the European places.
Everton secured an important victory over Blackpool last weekend in an enthralling 5-3 match and despite coming from behind to take all three points, Moyes admits he was concerned by his side's defensive vulnerability.
He told reporters: "I like to score - I am a supporter as well - I like to see goals in action.
"But when you are the manager and do the work - I look at the defensive mistakes we made in the last game and I can't take pride in that.
"Yes I scored five, but I conceded three so I have to do something about the bit that didn't go well.
"Scoring the goals was great but the bit that didn't go well was the conceding.
"Over the years we have been renowned for keeping it tight defensively."
The victory over Blackpool handed Everton an important three points and Moyes is hoping the result will act as the catalyst to belatedly kick-start the Toffees' season.
He continued: "We have got to hope we can go on a run, win three or four games on a row, and hopefully we'll be on the tails of the teams just outside Europe.
"We're only just into February so we've got to hope the best is still to come.
"In the last few years we've tended to do that and that gives the players a bit of belief that the form will come.”
He added: "It's coming a bit later than I'd like it to but hopefully we can come through now and get on a run."
Everton face a trip to the Reebok Stadium on Sunday to face Bolton Wanderers and Moyes accepts it will be a difficult game for his side to get a positive result.
He will be forced into making changes from the side that beat Blackpool, with Louis Saha a notable absentee after picking up an injury in the week.
And the Scot has called for his players to take the momentum from their last game into the match at Bolton, but admits his side are in for a tough game.
He said: "Injuries happen, though, don't they? You just have to get on with it. I'd rather talk about the players that are fit.
"I hope the players who are fit are looking forward to going to Bolton, on the back of a good result.
"But Bolton have done well this year. Owen Coyle has got his team playing very well and we know it is going to be a hard game."