IIRC The Persians and the Greeks were weakened by 1500 years of fighting each other and both fell victim to the Islamic armies from Arabia.
Persia had been fighting the Romans for ~1000 years or so. By the 7th century the Greeks and the Romans were essentially one, (Constantinople). There were also plagues in that time which surely took their toll. I need to read up on that. Wm. McNeill may have said something in Plagues and Peoples, which I could stand to re-read.