The Patriotic Front claims the poll was marred by discrepancies between vote tallies and the number of voters on registration lists.
Sata, who is refusing to accept the result, had led the vote throughout two days of counting with Banda emerging as leader late on Saturday.
Banda, acting president and leader of the ruling Movement for Multiparty Democracy (MMD), narrowly won power after securing around 40 per cent of the vote, compared with Sata's 38 per cent - a difference of just 35,000 votes. Turnout was about 45 per cent.
Scores of Sata supporters rioted in the capital, Lusaka, late on Saturday in protest at the result.
Witnesses said market stalls were torched and stones thrown in a slum area of the city. Police used tear gas to disperse the crowd.
The Patriotic Front has written to Zambia's election commission requesting a recount in 78 of the country's 150 constituencies. It also wants the recount to be verified by independent monitors and party agents, Given Lubinda, a party spokesman, said.
"We hope the recount and verification can be done soon ... we merely want a recount which will show that the results were not accurate," he said.
Before any ballots were cast, Sata warned he would not accept a defeat and accused election officials of rigging the poll and inflating the voter roll in favour of the ruling party.
The election commission has denied the charges, and African observers declared the vote free and fair.
Zambia's biggest monitoring group, the Foundation for Democratic Process, said its vote count tallied with official results.
"The findings should assure Zambians and political contestants that the official results announced by the electoral commission of Zambia reflect the will of the people," it said in a statement.
The election on October 30 was triggered by the death of the MMD's Levy Mwanawasa, who died in August, two years into his second five-year presidential term.
If Sata's legal challenge is unsuccessful, Banda will now serve the remaining three years of Mwanawasa's term.