Eyewitnesses said fighting began in the village of Qolosna in Upper Egypt's Minya Governorate after several Muslim taxi drivers reportedly bothered a group of Christian girls.
Shots were fired in the air and the injured - four Christians and three Muslims, according to security sources - were taken to hospital in the nearby town of Samalout.
Security sources said 10 people were arrested over the clashes and are being investigated.
Eyewitnesses said the military had imposed a local curfew starting after Friday afternoon prayers.
Coptic Christians account for about 10 percent of Egypt's population and have been repeatedly targeted by Muslim extremists.
As recently as last week, hundreds of extremist Muslims surrounded a church in a village 7 kilometres south of Minya and threatened to kill the local priest, the Assyrian International News Agency reported. The extremists began targeting the church iin March after renovation work began, threatening to demolish the church.
Twenty-four people were killed and around 100 injured in a New Year's Eve bombing of a Coptic church in the northern Egyptian port city of Alexandria.