The incident took place on the Dhaka-Sylhet Highway in Mahmudabad around 8 am on Thursday, according to Bhairab Highway Police chief Mozammel Hossain.
The victims have been identified as Farukh Mia from Raipura, 'Mustakim' from Nilkhuthi, Shahjahan Mia from Paglabari, and Mahmudabad-natives Ripon Mia and Bacchu Mia.
The covered van was heading to Sylhet when it suddenly went off the road and drove into a market in Mahmudabad, according to the police.
"A few people were buying groceries at the time. Two of them were killed on the spot," said Mozammel. Three others passed away on the way to hospitals in Bhairab and Dhaka.
The early indications are that the driver fell asleep behind the wheel, according to Mozammel. Police have seized the vehicle.