The deputy governor of Zahedan was quoted by Tasnim news agency as saying both explosions were caused by percussion grenades.
There was no immediate claim of responsibility for the blasts.
Zahedan is the capital of Sistan-Baluchestan province where Iranian security forces often clash with armed drugs smugglers and Sunni militants. The province is home to mostly Sunni ethnic Baluchis, a minority in predominantly Shi’ite Iran.
The Young Journalist Club, a news website linked to state broadcaster IRIB, quoted the head of Sistan-Baluchestan police Mohammad Ghanbari as saying "a suspicious package" was found on the street and exploded before being neutralized by a bomb disposal unit, injuring three police officers.