News of the battle came after two suicide car bombers targeted ethnic Tuareg forces in the northern town of Tessalit, killing three people.
A US official meanwhile confirmed that Washington had deployed several Predator drones to Niger to fly surveillance missions in support of French forces in Mali.
The clash between Chadian forces and the Islamists took place Friday in the mountainous Ifoghas region of northern Mali, Chad's military command announced.
"The Chadian army destroyed five vehicles and killed 65 jihadists," it said in a statement. But 13 of its soldiers had been killed and another five wounded, it added.
Earlier this month Chad deployed 1,800 soldiers in the northern city of Kidal to secure what had been the rebels' last urban stronghold, putting itself in the frontline in the fight against the Islamists.
A car bomb attack Thursday there near a military camp for French and Chadian troops had wounded two civilians.
North of Kidal lie the Ifoghas highlands -- to where many of the Islamist forces have withdrawn -- and the town of Tessalit.
Two vehicles targeting civilians and members of the ethnic Tuareg rebel group, the MNLA, exploded near the town Friday, killing three and wounding several others, a security source said.
A spokesman for the National Movement for the Liberation of Azawad (MNLA) in Burkina Faso confirmed the report.
A US defence official, speaking on condition of anonymity, meanwhile confirmed that drones were now flying out of Niamey, Niger, from a base with a 100-strong contingent of air force personnel.
President Barack Obama had announced earlier Friday that US troops had been sent there to provide intelligence for the French forces and "other partners" in the region.
France sent in troops on January 11 to help the Malian army oust Islamist militants who last year captured the desert north of the country. Since then, thousands of soldiers from African countries have also deployed and France plans to start withdrawing its troops next month.
The French-led forces met little resistance during the initial offensive that drove the Islamists from the main northern centres of Gao, Kidal and Timbuktu.
Now however, they are facing a guerrilla campaign that includes sudden raids, suicide attacks and land mines.
Rebels determined to retake Gao
The Tuareg MNLA blamed Friday's car bomb attacks on the Al-Qaeda-linked Movement for Oneness and Jihad in West Africa (MUJAO), one of Mali's main Islamist groups.
The MUJAO made no comment on the latest attacks, but on Thursday it told AFP it was responsible for the car bomb in Kidal. MUJAO spokesman Abu Walid Sahraoui warned of more such attacks.
Sahraoui also said they had sent fighters to another northern town, Gao, 1,200 kilometres (745 miles) from the capital Bamako and that they were determined to recapture it.
Battles with French-backed troops erupted overnight Wednesday after about 40 Islamists infiltrated the city and there were fresh clashes on Friday.
Malian soldiers opened fire with rocket-propelled grenades on the city hall, killing at least one Islamist, according to Malian army colonel Mamadou Samake.
They found the bodies of four rebels inside, he added.
A Malian soldier earlier said many of the Islamists killed in the urban clashes wore explosives belts, adding that mines had been laid in the area, French mine clearers had been called in to make the area safe.
The French military reported Friday that between 15 and 20 Islamists had died in the street clashes. Two French troops were slightly injured, while four Malian soldiers were also thought to have been wounded.
The International Committee of the Red Cross said tens of thousands of people who had fled northern Mali were still too scared to return, many fearing reprisals from Malian troops.
The UN said Friday that it had "heard horrifying reports from the north of human rights violations, recruitment of children and rising sexual violence".
Human Rights Watch has already urged Bamako to act on reports that Malian troops had targeted light-skinned people, particularly Arabs and Tuaregs for arbitrary arrest, torture and sometimes killings.
Ethnic Tuaregs in the north, who have long sought greater autonomy, initially backed the rebellion but later fell out with the Islamists and regained control of Kidal before the arrival of French troops.