The authorities in Argentina, Chile, Colombia and Spain carried out the arrests and seized 250 items of IT equipment and mobile phones, Interpol says.
Those arrested are aged between 17 and 40.
The operation follows cyber attacks on Colombian and Chilean websites.
Interpol says the websites of the Colombian Ministry of Defence and the presidency, Chile's national library, and the Chilean electricity company Endesa were attacked.
Spanish authorities say four people were arrested as part of the operation in Spain, in Madrid and Malaga.
Two of them remain in custody, while the other two, including a minor, were released on bail.
Spain's interior ministry said that one of the four arrested was the alleged manager of Anonymous' computer operations in Spain and Latin America.
Anonymous is a loose international network of online activists which describes itself as an "internet gathering". Its protests often take the form of disrupting websites and services.
Many Anonymous protests tackle issues of free speech and preserving the openness of the net.
Police in Latin America and Europe have arrested 25 suspected members of the Anonymous hacking group, according to Interpol.