"The tragedy, which was caused by explosions from the fireworks often used in these places, first consumed the building's ceiling, resulting in two very loud explosions, causing panic and a stampede," the ministry said.
"When we arrived it was a panic, there was a strong fire with lots of smoke," an official with the firefighters told AFP on condition of anonymity.
"We counted 16 dead and five injured," the official added.
A security guard present for the fire said that "it happened very quickly".
"It was a little after 2:00am and most customers arrive around 3:00am... there are many victims," the security guard said.
There were some burnt objects out the front of the club that suggested a fire, but the building's facade was not destroyed or charred, an AFP journalist said.
Cameroon is hosting the AFCON tournament despite regular violence in the country's west, where English-speaking militants declared independence from the majority French-speaking country in 2017.
Both the separatists and government forces have been accused of atrocities in the fighting, which has claimed more than 3,000 lives and forced over 700,000 to flee their homes.