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Mugabe, 89, has said he would step down if he loses. Many Zimbabweans find it hard to believe that the wily politician, a former guerrilla who led the country to independence in 1980 and has the backing of the security forces, would relinquish control even if the vote doesn't go his way.
After voting in Harare's western Highfield township, Mugabe said he expected to be re-elected.
"We will have lots of things to do to get our economy back on its feet," he said.
Tsvangirai, who had been in an uneasy power-sharing arrangement with Mugabe, cast his ballot in northern Harare.
"We have come to complete the change we have always fought for," he said. "It is an emotional moment for me but I am filled with a sense of calmness."
The coalition between the two rivals was forged by regional leaders after Zimbabwe's last disputed election in 2008.
Zimbabwe's shaky government was effectively dissolved on Wednesday; Mugabe and Tsvangirai have each predicted outright victory that would avoid the formation of another coalition.
Half the population of 12.9 million is eligible to vote. The state election body has said administrative, logistical and funding problems hindered voting arrangements, but said the problems have been fixed at the more than 9,000 polling stations nationwide.
Previous elections in 2002 and 2008 were marred by allegations of vote-rigging and political violence. Rights groups say there has been little overt violence this time but noted deep concerns over shambolic voters' lists, the role of Mugabe's loyalist police and military in the voting process and bias in the dominant state media and the sole national broadcaster controlled by Mugabe loyalists.
The International Crisis Group, a research organization, said it fears a return to a protracted political crisis and possibly extensive violence if the Zimbabwe poll is inconclusive and disputed.
Wednesday was a public holiday because of the vote. In Mbare township, the lines were moving smoothly with ballots being cast at an average of five minutes for each voter in the three votes for president, lawmakers and rural and urban council officials.
The election is being watched by observers from the regional Southern African Development Community, or SADC, and the continent-wide African Union. Some Western embassies in Zimbabwe have been permitted to deploy a limited number of diplomats to key voting districts.
Many voting lines were dominated by young voters and women. Rural voters in Mugabe's traditional strongholds also formed long voting lines, witnesses said.