However, the boss has confessed that his side's performance was below par.
"Tamworth did really well and kept at it and whilst it was only 1-0 they were in with a chance," said Moyes.
"There was threat from corners and free kicks but in truth I didn't think there was much and without us playing well I thought we had a level of control to see the game.
"Before today's game you're always nervous, worried about a banana skin, but I've got to praise the players for their attitude. We didn't play well but our attitude was very good."
The Toffees last won the FA Cup in 1995.