Hart is on loan at struggling West Ham and has failed to live up to the heights expected from him so far this campaign, having shipped 30 goals in his 14 Premier League appearances to date.
Four of those came against Everton in midweek and Moyes admits that the amount of work required of Hart and the pressure of scrapping at the bottom of the league will be new to the England No. 1.
"Let's be honest, he knows he is going to get plenty of work here at the moment, with the way we're defending," said Moyes. "He's getting that just now which means he will be under more pressure than perhaps he has been used to.
"He's done really well in some of the games so far. We've needed him for his leadership, his organisation. He's a man for us in the dressing room. These things are also important.
"He's someone who is always out trying to help the team."
Hart was blamed for Everton's third goal on Wednesday as his limp kick fell straight to the feet of Wayne Rooney, though the veteran striker still had plenty to do himself.
Despite that error, Moyes says that he is pleased with Hart's overall form, saying: "I think Joe's some done some really good things. He made a brilliant save against Watford down to his right.
"I was disappointed the other night. I told him his kick (which led to Rooney completing his hat trick) was poor. It should have either gone into the stand or away somewhere else.
"I think there's been mixed bits from Joe. I couldn't turn around and say that it has all been great. If you can get the jersey then you have always got a chance."