The bombing was the first incident on the Chadian side of the border since the Sudanese military started air attacks on positions of rebels who have been fighting Khartoum since February 2003.
A source close to the Chadian army said 10 civilians were killed and 16 injured in the bombing at 0900 GMT on Thursday.
However the regional coordinator for the UN High Commissioner for Refugees in Tine, Yvan Sturm, said two civilians were killed and 15 injured in the attack.
The deaths came as the UN High Commissioner for Human Rights called for an independent commission of inquiry to assess reports of "systematic" human rights abuses in the region.
The UNHCR official said two of the wounded were in a critical condition and were receiving care from the humanitarian group Medecins Sans Frontieres.
A MSF official said their hospital had received 15 people wounded in the bombing.
Around 100,000 people have fled
the fighting to Chad
"Most of the injured we have received so far are women or children," said Sonia Peyrassol, MSF field coordinator in Tine.
"Two of the heavily wounded are elderly. All are clearly civilians."
Some of the bombs hit near the dry riverbed which marks the border, along which many refugees had been seeking shelter.
The bombing caused panic among the refugees gathered in Tine and many fled further into Chad, away from the Sudanese border, said MSF.
Rebels in Darfur launched the rebellion to protest against alleged government neglect of the semi-desert region in the west of the country.
Clashes have intensified since peace talks in Chad collapsed on 16 December.
The rebellion has cost 3000 lives while another 100,000 people have been forced to flee into the east of Chad. Some 670,000 have been displaced within Sudan itself.
Sturm said the bombs dropped on Thursday hit the so-called Red Hill sector of Tine, where 1045 Sudanese refugees have been sheltering since fleeing the conflict in Darfur.
Chadian military sources said late on Thursday that the Sudanese army took control of the Sudanese side of Tine from the rebel Movement for Justice and Equality in Sudan (MJE) during the afternoon.
Darfur rebels say Khartoum is
neglecting the region
The Sudanese side of Tine is devoid of inhabitants, who fled to escape the regular bombardemnts from government forces.
Meanwhile, United Nations acting High Commissioner for Human Rights Bertrand Ramcharan said he was "deeply concerned" by the deteriorating human rights situation in Darfur.
In a statement released in Geneva, he pointed to reports of "systematic human rights abuses against unarmed civilians", as well as the burning and looting of villages.
Ramcharan called on both the government in Khartoum and the rebels to "agree on the establishment of an independent international commission of inquiry to assess the humanitarian and human rights situation in Darfur".
He said he was also extremelely concerned by reports that pro-government militias like the Janjaweed have been "heavily involved" in the violence in the region.