Santos, an 18-year-old who already stands at 6'5", has penned a three-and-a-half-year deal with Posh.
"I still think I am growing," Santos, previously of Thurrock, told the club website.
"I am really pleased to be at Peterborough, I know a few of the lads already and the club is very ambitious."
Posh manager Darren Ferguson added: "He is one for the future, but one that we know that a lot of clubs were looking at. Potentially, we feel he is a good signing for the football club."
Peterborough are currently just outside the playoff places in League One.