Officials have said the attack is the biggest to hit the region in several months.
"It is the biggest attack so far in terms of number of injuries," said Colonel Parinya Chaidilok, a military spokesman.
He also said that 70 people were wounded, including five with serious injuries.
Lieutenant General Surachai Suebsuk, the Narathiwat police commander, said that one of five people seriously hurt - a woman who was an elected community representative - died later in hospital.
According to forensic police, about 30km of explosives had been packed into the car which was left in the fruit market parking lot and detonated by mobile phone.
The second blast occurred when about five kilograms of explosives concealed in a motorcycle were detonated, also by mobile phone.
"The bombs show that the wrongdoers want to attack places where there are many people," said Somchai Wongsawat, the Thai prime minister.
"The wrongdoers want to make trouble and keep the unrest going."
On Monday night, a 47-year-old religious teacher was shot dead in Narathiwat province, while a 41-year-old man was killed later the same night in a similar attack in nearby Pattani province, police said.
Narathiwat is one of three majority Muslim provinces in the far south of predominantly Buddhist Thailand, bordering Malaysia.
The region has seen scores of attacks over the past five years that police have attributed to Muslim separatist fighters.
More than 3,000 people have died.
The violence has ranged from drive-by shootings and bombings, to beheadings and appears to target both Buddhists and Muslims associated with the Thai state, such as police, soldiers, teachers and government officials.
Human rights groups also accuse the Thai military and police of carrying out atrocities.