Jordan Pickford saved a penalty from Newcastle winger Anthony Gordon to keep the clean sheet.
Dyche later said, "We have been patching teams together for a while now and we have asked a lot of Jimmy Garner and he did very well. I ask a lot of the players by trying to get them Premier League fit and injury free. That mentality continues to grow and it served us well last season.
"We carry a small squad for financial reasons and if you get injury and illness it affects you and stretches you."
On Everton's mentality, he also stated: "I think it is coming back. Last season we had a strong jaw against many things that were going against us. There have been mistakes this season and we have switched off in key moments. That is reducing now and we are doing the basics better and still trying to find as way of winning games.
"It is hard to correct everything and we are showing signs that we are."