Zarif made the remarks in a meeting with families of some Iranians detained in various countries on sanctions-related charges on Monday.
The families were also briefed on the latest status of the cases of their loved ones.
The foreign minister also described the detentions and requests of unreasonable extradition as illegal and politically motivated.
Zarif added: "The Foreign Ministry will use all its capacities to pursue the cases that are totally illegitimate and cruel, and are tantamount to hostage taking for ransom."
The United States has sanctioned Malaysia-based Mahan travel and tourism company, because of its links to Iranian airline, Mahan Air. The sanctions came into effect on Monday.
Mahan travel and tourism, based in Kuala Lumpur, Malaysia, has been subjected to secondary sanctions under Iranian financial sanctions, according to US treasury department's office of foreign assets control (OFAC).
Iranian President Hassan Rouhani at the meeting of top businessmen of Iran and Austria in Vienna said US sanctions create problems for people’s lives in Iran, and is 'an international crime'.
