The Irish defender, 20, was recalled from his loan at Scunthorpe at the start of the transfer window with first-choice defenders Phil Jagielka and Sylvain Distin injured.
He was initially recalled to provide cover, but made his Premier League debut off the bench at Tottenham and has followed that up with starts in the last two games.
"Shane has done really well and he has got something. He will certainly have a good career in the game and we hope it is at Everton," said Moyes.
"But the level is going to step up - the more you play, people start to look for more from you.
"We want to Shane to keep doing things simply, not over-complicating it. He will be feeling good about himself and we want him to play that way, but we will have to keep an eye on him closely."
Moyes must decide whether to retain Duffy for Friday evening's FA Cup fourth-round tie against Fulham at Goodison Park or reshuffle his back line by recalling fit-again veteran Tony Hibbert.