Robinson, 35, is a former assistant at the Roosters, and has been in charge at the Dragons for the past two seasons.
The 2011 Super League Coach of the Year will replace Brian Smith, who was this week sacked by the Roosters.
Robinson said: “I would like to thank Bernard Guasch, the club, players and supporters for the two years I have coached at the Dragons. The progression that we have as a club has permitted me the opportunity to be offered the highest honour as a coach in the NRL.
“Thanks again to the players for the belief in the direction we undertook two years ago, Bernard and the Board of Directors and Christophe Jouffret to back us in our vision and the supporters for their incredible force on pushing us to achieve and surpass our potential.”
Catalan finished bottom of the table the season before Robinson’s arrival, and he has transformed them in to a major force in the top flight, finishing in the play-offs in both of his seasons in charge.