How did Sasanian Persia and Rome fight for four centuries, and who won?

Nobody won. The Sasanians and the Romans, later the Byzantines, fought off and on for centuries, and in the end both empires were so exhausted that a third power swept in. The Sasanian Empire was an Iranian empire founded and ruled by the House of Sasan from 224 to 651 AD, and its reign, second only to that of the Parthian Arsacids, lasted over four centuries.

It began in 224, when the Sasanians defeated the last Parthian king. They then competed far more zealously with the Roman Empire than the Arsacids had, starting a new phase of the wars. In 260, at Edessa, the Roman army under Valerian was defeated and captured, and for the first time a Roman emperor was taken prisoner.

Most of the time, though, the border barely moved. The conflict was a tug of war, because neither side had the logistics or manpower to campaign far from home. The Roman–Persian wars lasted nearly seven centuries and ended in an inconclusive outcome.

The last round came in 602, when Khosrow II used the murder of the Byzantine Emperor Maurice as a pretext for a new invasion. Jerusalem fell in 614, Alexandria in 619, and the rest of Egypt by 621. But the campaign had exhausted the Persian army and treasuries. In 628, Khosrow was overthrown by his son Kavadh II, who ended the war.

By then both sides were drastically exhausted, and the Sasanians were weakened by social conflict. They were vulnerable to the sudden emergence of the Rashidun Caliphate, whose forces invaded both empires only a few years after the war.