Medusa's head continue to have power after it was detached from her body by Perseus. In fact, upon beheading Medusa, two offspring came out from the drops of her blood - Pegasus, a winged horse and Chrysaor, a giant or a winged boar.
Perseus also used Medusa's head as weapon in most of the fights he had and later on requested to place Medusa's head into his shield so that everyone who will target him shall turn into stone.
Perseus used the "Gorgon's Head" to turn Polydectes into a STATUE.
This was just a hunch.
If you looked at Medusa's eyes, you will be a stone. The head still had her eyes, so Perseus benefits from it as a sign of defense.