The men were eventually convicted but appealed the judgement on the grounds that Rahman's death was part of a mutiny and they should have been tried under martial law instead of through the civilian court system.
The five convicted men were in Dhaka central jail when the jury dismissed their appeals, but crowds gathered outside the court ahead of the ruling.
"There are scenes of joy outside the court here, as people are celebrating the court ruling," Nicholas Haque, Al Jazeera's correspondent in Bangladesh, reported.
Police were on high alert and security was also being provided to lawyers and judges in the case, Nur Mohammad, the national police chief, said.
Rahman led Bangladesh's independence war against Pakistan in 1971, until which time Bangladesh was considered as eastern Pakistan.
Rahman, along with most of his family and close aides, were gunned down at his Dhaka residence on August 15, 1975, during a military coup.
Sheikh Hasina and her younger sister, Rehana, survived the coup because they were touring Europe at the time.
Originally, 15 men - mostly former army officers - were sentenced to death for Rahman's killing, but in 1998 Bangladesh's high court reprieved three of them.
Of the remaining 12, six fled and were never caught, one died abroad.
Legal experts said the men on death row would still have the right to apply to the supreme court for a review of Thursday's verdict and then finally to the president for mercy.