Silva, 18, came through the club's academy and has impressed at the start of pre-season.
He said: "This is amazing. Pre-season has gone really well, as Alex said I need to keep working hard and doing what I've done to get here."
Silva scored the equaliser at Harrowby and a hat-trick against Bourne Town, as well as winning penalties at Newark Flowserve and Cotgrave.
Manager Paul Rawden said: "Seeing his attitude to start with is brilliant he's young and hungry. He's learning from the older heads around him. We saw more than enough to know he's a real talent."
Alex Watson, Academy Director, added: "To be the first Academy player to be awarded a contract with the first team is very special.
'It's been really encouraging to see Academy players taking part in the first team training, Paul and Russ are really keen to have that pathway.
"Rodrigo is a great lad and hopefully he will go far, but in the meantime he has to get his head down with the first team and carry on how he's started."
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