Zanu-PF party says the scale of the violence has been exaggerated and accuses the MDC of staging political attacks.
Chief Elections Officer Lovemore Sekeramayi said former Finance Minister Simba Makoni came third with 8.3%.
He said a date for the second round between Mr Mugabe and Mr Tsvangirai would be announced later. It should be held within three weeks.
Mr Makoni held back from endorsing either candidate after the results announcement, saying instead Zimbabwe could not afford another round of voting, the AFP news agency reports.
"The way forward for this country is for the political leaders to work together," he said.
A spokesman for Mr Mugabe's Zanu-PF, Bright Matonga, told the BBC that both main parties had been aware that no-one had won outright.
Chief elections officer announces the results
"The laws of Zimbabwe and the constitution clearly states that for one to be an outright winner, they have to achieve 50 [per cent] plus one," he said.
"We are following our constitution, not people's wishes."
MDC Secretary General Tendai Biti told the BBC that Mr Tsvangirai should be declared president, as he had gained 50.3% of the vote.
Mr Tsvangirai has in the past said he would not take part in a run-off unless international observers were involved.
He has moved out of Zimbabwe amid fears for his safety.
Before the official results were announced, the MDC had queried 120,000 votes given to Mr Mugabe - which they say could have tipped the balance in the president's favour and stopped Mr Tsvangirai winning outright.
"This whole thing is a scandal, scandalous daylight robbery," said MDC spokesman Nelson Chamisa, after the results were announced.
"What we are being given here as the outcome are some fudged figures meant to save Mugabe and Zanu-PF."
But projections from independent monitors were similar to the official results, with Mr Tsvangirai just short of the threshold for outright victory.
The UK and United States have raised doubts about the results and the possibility of a second round being fair.
The UK Foreign Office said the results "lack credibility", adding that a second round would not be fair unless the violence ended and more international monitors were present.
Robert Mugabe came second in the vote but is determined to keep fighting
"President Mugabe's campaign of violence and intimidation coupled with the arrest of 99 electoral commission officials in the last month show exactly how Zanu-PF will approach any second round," a spokeswoman said.
The election officials have been accused of trying to rig the elections in favour of the MDC.
US state department spokesman Tom Casey said it was hard to see how a run-off could be fair "when the government has done everything it could to both delay and obscure the results".
The MDC and human rights groups say there has been a massive campaign of violence against opposition activists in rural areas in anticipation of a run-off vote.
The MDC says hundreds of people have fled their homes and 20 have been killed.
On Thursday, the defence minister in neighbouring Botswana said Zimbabweans were fleeing the violence, with almost 100 people arriving in the past three days.
He said in the past, Zimbabweans had been economic migrants but now they were seeking political asylum.
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