Riva, whose physical presence and powerful shooting earned him the nickname Rombo di Tuono (thunder-clap), scored 35 goals in 42 appearances for the national team.
Riva had been taken ill at his home in Sardinia over the weekend and treated for a suspected heart problem.
He was part of the Italy team that won the European Championship in 1968 and also played in the 1970 World Cup when they were runners-up to the great Brazil team featuring Pele.
"I am shaken and deeply saddened. Italian football is in mourning because a true national monument has left us," FIGC president Gabriele Gravina said.
The FIGC statement added a minute's silence in his memory will be observed before all Italian championship matches this weekend.
With his near namesake Gianni Rivera, Riva was one of the poster boys of Italian football as colour television brought live action to a wider audience. He was runner-up to Rivera for the Ballon d'Or trophy awarded to Europe's best player in 1969.
Riva played almost his entire club career with Cagliari and was Serie A top scorer when the Sardinian team won the Italian league title for the only time in their history in 1970. He was honorary president of the club at the time of his death.
Injuries marred the latter part of his playing career and forced him to retire in 1976, having scored 164 league goals for Cagliari in 315 matches. He was Serie A top scorer three times.
Riva subsequently served for many years as a member of the backroom staff for the national team including in 2006 when Italy won the World Cup.
