In a speech that followed, Mr Mugabe said he was committed to talks with the opposition to find a solution to the political crisis.
"It is my hope that sooner rather than later, we shall, as diverse political parties, hold consultation towards... dialogue as we minimise our differences and enhance the area of unity and co-operation," he said.
But Mr Tsvangirai told the BBC's Focus on Africa programme: "We believe the 29 March election reflected the will of the people.
"That should be the basis upon which any negotiation... for a transition should be based."
Archbishop Desmond Tutu speaking to Andrew Marr
Electoral commission officials said voter turnout had been about 42%, similar to that of the disputed first-round vote.
Independent observers said many of those who did go to the polls voted out of fear, and that thousands of people had spoiled their ballots by defacing them or marking them inaccurately.
President Mugabe is now expected to fly to Egypt to attend an African Union summit which opens on Monday.
The BBC's Peter Biles, in Johannesburg, the question now is whether Zimbabwe's neighbours will recognise this election result.
An observer team from the Pan-African Parliament has called for fresh elections to be held, saying the vote was not free or fair.
It has urged Sadc and the African Union (AU) to facilitate talks between the government and opposition.
Earlier, the former Archbishop of Cape Town, Desmond Tutu, appealed to the AU not to recognise Robert Mugabe as president.
Archbishop Tutu told the BBC that Mr Mugabe should have no role in a future government, and said that he would support the deployment of a UN force to restore peace in the country.
There has also been international outrage at events in Zimbabwe.
US Secretary of State Condoleezza Rice said it was time for strong international action to end the violence in Zimbabwe.
Ms Rice, meeting her Chinese counterpart in Beijing, urged members of the UN Security Council to back proposals for an arms embargo and a travel ban on Mr Mugabe's associates.
It follows President George W Bush's instructions on Saturday to US officials to come up with new sanctions against Zimbabwe.
The MDC says some 86 of its supporters have been killed and 200,000 forced from their homes by militias loyal to the Zanu-PF.
The government blames the MDC for the violence.
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