The Bangladeshi election commission said it would formally announce the results on Wednesday.
Nuh Alam Lenin, a spokesman for the Awami League, said: "Our leader has called for change and the people have responded to her call.
"They have given a thumping verdict against corruption and criminalisation of the past regime," he said.
Hasina, who led Bangladesh from 1996 to 2001, has urged supporters to avoid public celebrations until the final results are announced.
"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.
However, the BNP said on Tuesday that its supporters were unable to vote in various parts of the country, and confirmed 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.
Zia also said on Tuesday there had been widespread vote rigging.
"We have confirmed reports of rigging and other irregularities in many polling stations across the country," she told reporters, giving her first reaction to the election.
"So we reject the election outcome," she said.
"We are collecting details of more irregularities and will give to the media and appropriate authorities over next few days," she told reporters at her Dhaka party office.
However, election officials and monitors said the polls were mostly peaceful.
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.
Monday's parliamentary vote was held in order to return Bangladesh - a country of more than 140 million people - to democracy after two years of emergency rule imposed by the military government.
The new government will have to tackle the corruption, widespread poverty and chronic political and social unrest which prompted the military to intervene in the first place.
Although jailed by the interim military government on charges of corruption, both Zia and Hasina campaigned on promises to fight endemic graft.
Bangladeshi political parties tend to be driven by personalities rather than ideology, and analysts who won Monday's election may be less important than the losers publicly accepting the result.