Midfielder Dias and striker Moyo join Lewis Hornby and Dean Snedker, who have already agreed deals for next season.
"Both David and Claudio have earned their deals but the hard work starts now," manager Aidy Boothroyd told the club's official website.
"In many ways, your first professional contract is the easy part. They must keep working hard, listening and gaining experience every day to become better players.
"Football is a very competitive industry and both players know what is ahead of them.
"I have been impressed with their performances on the pitch and their attitude off the pitch and I would like to congratulate them on their contracts."
- Sports & Recreation
- Aidy Boothroyd