The barge had been anchored off the port but dragged anchor because of the high winds and rough seas in recent days.
It ran aground near the colonial-era fort, about 20 feet from the shoreline.
The vessel's agents were scheduled to arrange for the barge to be pulled out to sea and towed back to its anchor position when the weather improves, port officials said.
Port officials said it was not unusual for small vessels like barges and fishing boats to drag anchor and get washed ashore in bad weather. Galle is a convenient port of call as it is close to the main East-West shipping lane just south of the island.