The Awami League, led by Hasina, ruled Bangladesh from 1996-2001.
Monday's parliamentary vote returned Bangladesh, a country of more than 140 million people, to democracy after two years of emergency rule imposed by an military-backed government.
While final results are still awaited, Sheikh Hasina's associates called for patience.
"Our leader Sheikh Hasina has appealed to her party and supporters not to stage victory marches or engage in any kind of celebration until the final results are announced by the election commission," Abul Kalam Azad, her spokesman said in a television broadcast.'Voter intimidation'
A leader of the BNP said on Tuesday that its supporters were kept from voting in various parts of the country, and it planned to file a complaint.
|Monitors say the vote went smoothly despite claims of voter intimidation [Reuters]
"We have reports that BNP supporters were barred from coming to the polls and also were driven away from polling stations in many places," Rizvi Ahmed said on local television.
Zia herself had said on Monday that if the election was fair, she would win.
However, election officials and monitors said the polls were mostly peaceful, and saw few glitches.
Previous elections have been marred by violence and widespread accusations of
"The election ended in a very peaceful environment and I never saw such a congenial atmosphere. The turnout was tremendous," Taleya Rehman, executive director of Democracy Watch, a monitoring group, told the Reuters news agency.
A military-backed interim government took control of Bangladesh in January 2007 amid widespread political violence and cancelled elections.
The winner of Monday's vote will have to tackle the corruption, widespread poverty and chronic political and social unrest which prompted the military to intervene.
Although both jailed by the interim military government on charges of corruption, Zia and Hasina both campaigned on promises to fight endemic graft.
Bangladeshi political parties tend to be driven by personalities rather than ideology, and analysts say it may be less important who won Monday's election than that the losers accept the results.