The deal's value could rise to a reported £10m as Chelsea owner Roman Abramovich continues to invest heavily.
Yet to play for Sao Paulo, Piazon is currently with the Brazil Under-17 side at the South American Championship.
Piazon had been linked with a move to Juventus before Chelsea made their move over a pre-contract deal, with talks still continuing.
Abramovich has already spent heavily on one Brazilian this season, signing defender David Luiz from Benfica for more than £20m in January, with Spaniard Fernando Torres also arriving at Stamford Bridge for £50m.
As well as Luiz, midfielder Ramires and defender Alex are the other Brazilians in Chelsea's first-team squad.
Sao Paulo, the club that produced Kaka, have already seen former players succeed in the Premier League, including Juninho at Middlesbrough.
Juliano Belletti was at the Brazilian club from 1996 to 2002 before playing for Chelsea from 2007 to 2010.