Iranian ambassador Ali Akbar Salehi signed an additional protocol to the nuclear Non-Proliferation Treaty at the Vienna headquarters of the UN nuclear watchdog, the International Atomic Energy Agency (IAEA).
The move was hailed as a watershed for Tehran as it tries to fend off accusations that it is seeking nuclear arms.
"This is an important step towards rebuilding international confidence in Iranian intentions regarding its nuclear programme," Straw said in a statement.
"But much remains to be done," he added. "It is now important for (Iran) to ratify the protocol as soon as possible, in order to give the international community the assurances it requires.
"And, as required by the IAEA, we need to see complete and verified suspension of Iran's enrichment and reprocessing activities".
Straw added: "Iran has already said that she will act as though bound by the additional protocol. I welcome the access already being granted to IAEA inspectors."