Mnangagwa has asked the country to let bygones be bygones, but many recall his role as state security minister during the Matabeleland massacres in the 1980s when the army killed thousands of people as Mugabe moved against a political rival. The MDC is headed by Nelson Chamisa, 40, a lawyer and pentecostal preacher.
Voting began at 0500 GMT (7am ) and polls will close at 1700 GMT (7pm). The result will be announced within five days, authorities said.
"The governing ZANU-PF party needs to maintain a semblance of free and fair elections in order to attract fresh foreign investment,"said the London-based EXX Africa business risk consultancy".
A recent Afrobarometer survey of 2,400 people put Mnangagwa on 40 percent and Chamisa on 37 percent, with 20 percent undecided.
Mnangagwa, a former spy-chief of Mugabe who took the presidency in 2017, is nicknamed "the crocodile" for his perceived ruthlessness.
There have been reports of intimidation and coercion and state media are biased towards ZANU-PF.
Zimbabwe's former leader, who appeared to have finally allowed his hair to go gray, spent the final question musing on his legacy, reminding people of his years in prison during the country's fight for liberation and his work raising literacy and education standards. "I was defeated at one time by Tsvangirai but he did not get enough votes to get an outright win", he said.
Zimbabwe's opposition leader Nelson Chamisa casts his vote.
Elections during Mr Mugabe's authoritarian rule were marred by fraud and violence, and this year's campaign has been dominated by accusations that the vote will be rigged.
"Zimbabwe's economy requires commercial bridge loans, donor support, relief on its huge dollar-denominated debt and, ultimately, an International Monetary Fund program", analysts at Exotix said in a research note on Monday.
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Some voters were being turned away at polling stations as their names were not captured on the voters roll, even though they claimed to have registered and received registration slips.
The current president is a man called Emmerson Mnangagwa, who used to be the vice-president and is now in charge of the Zanu-PF party.
Mnangagwa replaced Robert Mugabe in November previous year following a much-publicised military intervention. "The place has changed", said Patrick Musemburi, a 37-year-old minibus taxi driver in the capital Harare.
While voting proceeded smoothly in some areas, in others it was "totally disorganized", and it was unclear whether this was a coincidence or bad organization, he said.
President Mnangagwa urged voters to be peaceful on Monday.
"We are very happy to have a different vote", he said.
He added that: "We had that freedom after 1980".
Whoever wins, we wish him well ... In a surprise intervention on Sunday, Mugabe said he would vote for the opposition, turning on his one-time allies.
"I hope the choice of voting tomorrow will throw, thrust away the military government and bring us back to constitutionality", said a frail looking Mugabe, in a rambling off the cuff speech that lasted nearly an hour.
"I must say clearly I can not vote for those who have tormented me".