A female school cleaner was killed and a schoolgirl required an operation for her injuries in the strike on the town of Kilis just north of the border with Syria, the local governor's office said.
The town was hit by a total of three rockets, two of which fell into an open area while one more hit the yard, it said.
The rocket fire "came from Syria", it added, without giving further details. Reports said two more were lightly wounded.
Earlier reports had described the fire as mortar shells but later Turkish media reports described the objects as Katyusha-type rockets.
Television pictures showed the casualties being taken to hospital by ambulance. Windows on the ground floor of the school had been smashed by the impact of the blast while one car was severely damaged.
"The people should not allow provocations. Kilis residents should stay calm," Kilis mayor Hasan Kara told NTV television.
'IS-controlled area'
Kilis, a town of just under 100,000, lies north of the Syrian border, some 10 kilometres (six miles) from the Syrian town of Azaz.
Turkish officials have said it is the only town in Turkey with a majority of Syrians, some of the estimated 2.2 million living in Turkey who have fled the civil war at home.
That area of northern Syria south of Kilis has in the last months been the scene of fierce clashes involving Islamic State (IS) jihadists who have seized swathes of northern Syria.
The Hurriyet daily said that the army had found the fire came from an area in Syria controlled by IS and the armed forces had already returned fire on IS-controlled positions.
According to the Anatolia news agency, army radar showed the rockets came from a distance of 20 kilometres (12.5 miles) away and Turkish forces were able to destroy the launch positions with their fire.
Turkey has said that it is working to push IS out of the border zone, which the extremist group controlled on the Syrian side for much of 2015.
A suicide bomber who the Turkish authorities say was a member of IS killed 10 German tourists last week in the centre of Istanbul, the first time that foreigners have been targeted by such an attack in Turkey.
Turkish ground forces then pounded some 500 positions of IS in Syria and Iraq with artillery and tank fire over a 48-hour period, Prime Minister Ahmet Davutoglu said.
'Push IS from border'
Turkey has often been criticised by its Western allies for not doing enough to combat IS jihadists.
But Ankara last year stepped up its involvement in the US-led coalition against IS, hosting American war planes at its Incirlik air base for deadly raids against the jihadists and conducting air strikes of its own.
It is also relying on pro-Ankara Turkmen Syrian fighters who are on the ground in northwestern Syria to fight both IS jihadists and the regime of President Bashar al-Assad.
Turkish media reports have said Turkish jets are no longer flying over Syria after the shooting down of a Russian war plane on November 24 by one of Turkey's jets led to an unprecedented crisis in relations between Moscow and Ankara.
Turkey has also repeatedly said it does not want Kurdish militia forces -- who are themselves bitterly opposed to IS -- advancing westward of the Euphrates River in northern Syria.
Davutoglu said last week that Turkey would carry on fighting IS "until it leaves the Turkish border area completely."
