The convoy was attacked while travelling between the Menaka and Indelimane regions on Thursday, the fifth anniversary of the launch of France's military campaign against jihadists in the west African country.
“A vehicle approached the convoy and exploded near a VAB (armoured vehicle),” French army spokesman Patrick Steiger told AFP.
The seriously injured soldier was evacuated to France, while the two others were treated at the scene, he said.
Three Malian soldiers were also injured on Thursday after being ambushed in the northern town of Hombori in an attack using “improvised explosive devices”, Mali's army said in a statement.
Also on Thursday, a policeman was abducted and a police station looted and set on fire in the northwest town of Lere, the army said in a separate statement.
Islamic extremists linked to Al-Qaeda took control of the desert north of Mali in early 2012, but were largely driven out in the French-led military operation launched in January 2013, but large tracts of the country remain lawless.
According to an AFP count, 18 French soldiers have been killed in Mali since the launch of Operation Serval, which has since been replaced by Operation Barkhane, a broader offensive deployed in five countries — Mali, Mauritania, Niger, Chad and Burkina Faso.
These countries form the so-called G5 Sahel, a French-supported group that last year launched a joint military force to combat jihadism.