The Holland international will arrive at Stamford Bridge in the summer after seeing out the current campaign with PSV.
A routine medical was passed on Monday, and the deal is further evidence of owner Roman Abramovich's determination to bring the cream of Europe's talent to West London.
Manchester United had been interested in signing Robben but their bid for the former Groningen man was well below what PSV chairman Harry van Raaij anticipated and Chelsea's greater willingness to meet the asking price swung the pendulum in their favour.
The 20-year-old has four caps for his country and is reckoned to be a star in the making.
It increases the number of options within the Chelsea squad for next season and sees another top talent from the continent lined up for The Premiership.
