Army deserters allied to a rebel group called the Rally for Democracy and Liberty (RDL) mounted the early morning attack in Adre, a few kilometres from the border on Sunday, said Hourmadji Moussa Doumgor, Chad's communication minister.
Chad accuses the RDL of being a "militia used by the Sudanese government".
"The Chadian government holds the Sudanese government totally responsible for this morning's attack mounted from its territory," Doumgor said in a statement.
Sudan's Foreign Ministry denied involvement in the attack.
"Sudan is not planning or doing anything against Chad," said Jamal Ibrahim, a ministry spokesman.
Chadian forces were still pursuing rebel elements on Chadian soil but would chase them over the frontier into Sudan if necessary, Doumgor said.
"Sudan is not planning or doing anything against Chad"
Sudanese Foreign Ministry spokesman
"That would be justified," he said.
Doumgor said earlier that losses on the rebel side had been worse than on the government side.
The toll could not immediately be independently verified.
Aid workers on the Sudanese side of the border said the attack was the worst offensive in an escalating conflict.
Earlier Sudanese rebels and aid workers reported hearing large explosions and heavy fighting near Adre.
Scores of Chadian soldiers deserted their barracks in late September before regrouping near the border, and the government has accused Sudan of using the deserters to fight rebels in Darfur and of backing Chadian rebel activities.
Army deserters have demanded
Idriss Deby's resignation
Sudanese army sources reported sporadic fighting in recent days, crossing over the long, porous border between the countries, but added the Sudanese army was not involved.
Both Darfur rebels and aid workers in the region have reported large troop movements over the past two weeks near the border, with reports of Chadian troops patrolling on the
Sudanese side of the border.
The deserters have demanded President Idriss Deby's resignation. They are also accused of attacks on army bases in the capital, N'Djamena.
The clashes add to tensions in Darfur, which has been in revolt for almost three years. One of the main Darfur rebel tribes, the Zaghawa, lives on both sides of the border.
A Sudanese army source said on Sunday that Sudanese forces were in control of all the towns and villages but could not prevent the clashes spilling over the border.
"We are in total control of the villages and towns. All these clashes are happening outside the villages on open territory," he said.
A number of Darfur rebel commanders fought in the uprising that brought Deby, a member of the Zaghawa tribe, to power.
That revolt was launched from Sudan's remote west, a vast area the size of France.