Final answer:
Umar ibn al-Khattab, the second Rashidun Caliph, was the emperor during the Islamic conquests of the Sasanian Empire, overseeing significant territorial expansions and the downfall of the Persian power.
Step-by-step explanation:
The emperor who reigned during the Islamic conquests of the Sasanian Empire was Umar ibn al-Khattab, the second of the four 'rightly guided' or Rashidun Caliphs. After the death of the Prophet Muhammad, the Islamic state rapidly expanded under the rule of the Rashidun Caliphs, including Abu Bakr, Umar, Uthman, and Ali. The significant territorial gains against the Sasanian Empire were made during the caliphate of Umar (r. 634 - 644 CE). Under his leadership, the Arab-Muslim armies conquered large swathes of territory, which marked the decline of the Sasanian Empire and the spread of Islam.
During Umar's reign, conquests were utilized as a means to unify the Arab tribes and expand the faith. The military campaigns against the Sasanian Empire were effective, and by the time of Umar's assassination in 644 CE, significant portions of the empire had been brought under Islamic rule. It was also during this period that the Persian ruler Yazdegerd III fled to escape capture, and the Arab-Muslims controlled the entirety of the Persian Empire by 651 CE.