The coalition led by Hasina won more than two-thirds of the 300 parliament seats in the election, Bangladesh's first in seven years. Zia's coalition took just 31 seats.
Despite statements by foreign and local election observers that the election was free and fair, Zia said the results did not reflect the will of Bangladeshi voters.
"It is a farcical election," she said in a televised news conference on Wednesday.
In turn, Hasina has urged Zia to accept the result, describing the outcome as a vote for "governance over misrule".
Hasina said she was prepared to offer the opposition the deputy speaker post among others in the legislature.
Ministerial positions could also go to the opposition if it was willing to cooperate with the government, she said.
The vote dispute came after reports of some post-election violence on Tuesday during which one man was killed and around 150 injured in clashes between rival political activists, the private Diganta television station reported.
The main clashes occured in the northern Pabna area, the station said, with outbreaks of violence reported in other parts of the country as well.
However, local and international observers expressed satisfaction with the voting process.
Monitors from Commonwealth nations issued a statement saying the election was "credible... and the tabulation and results processes were transparent".
The statement concluded that "most key democratic benchmarks" had been met.
Despite this, BNP officials filed complaints with the election commission, alleging ballot-rigging and forgery at 220 polling stations, including instances where officials are accused of registering fake votes.
|Election observers say the vote was free and fair despite claims of fraud [Reuters]
Bangladesh's election commission said it would investigate.
More than 81 million eligible voters were photographed and given national ID cards in a process that helped eliminate about 10 million fake or duplicate names.
However, attempts to fight corruption have produced mixed results, including the failure to prevent Hasina and Zia from competing in the election.
Both were among around 200 top politicians charged with corruption, but they were later freed by the country's high court before the vote.
Charges were subsequently dropped against Zia, although her two sons - who also faced corruption charges relating to her 2001-2006 premiership - were forced to leave to country as part of a deal allowing her to stand in Monday's election.
Corruption and murder charges against Hasina have been dropped or frozen, although, in theory at least, she could still face some of the charges.
The murder charge relates to the death of four protesters who were killed during pre-election violence in 2006. That election was eventually cancelled by the military-backed government.
Hasina and Zia have dominated Bangladeshi politics for two decades, a state of affairs which analysts say is more a reflection on South Asia's penchant for political dynasties than the role of women in this impoverished nation.
Though bitter rivals, their parties campaigned on similar platforms of reducing corruption and controlling inflation.
Critics say that while in office, the BNP and the Awami League both failed to significantly alleviate conditions for the 45 per cent of Bangladeshis who live below the poverty line.