Mnangagwa is the frontrunner but the latest poll shows a tight race.
A run-off vote is scheduled for September 8 if no presidential candidate wins at least 50 percent. Younger voters are expected to be key - nearly half of those registered are under the age of 35. "I don't think the electoral commission is captured‚ no‚" he said.
Parliamentary and local elections are also taking place on Monday.
Chamisa has vowed not to boycott the vote, saying his party would still win despite accusing Mnangagwa and the Zimbabwe Electoral Commission (ZEC) of trying to fix the result.
He tweeted that Zimbabweans should vote with "peace in our hearts".
There were concerns in urban areas, which are opposition strongholds, that long queues were moving too slowly, raising fears of a repeat of the 2002 disputed elections that saw polling stations closing before thousands of people could vote.
Chamisa was previously a student political leader and protege of Tsvangirai.
Only 350 voters were expected to cast their votes at that particular polling station.
A contested result is likely to lead to street protests and possible violence.
The two main presidential rivals voted Monday morning, and each said they were confident of victory.
Mugabe ruled the country for almost four decades in a period marred by economic crises and corruption.
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But he has lived in obscurity since he stepped down under military pressure in November.
Mr Mugabe made no comment to reporters as he cast his ballot around lunchtime accompanied by his wife, Grace.
A huge crowd gathered outside, some cheering, many booing.
Mnangagwa, a former spy-chief of Mugabe who took the presidency in 2017, is nicknamed "the crocodile" for his perceived ruthlessness.
"I can assure you that this country is enjoying democratic space which has never been experienced before", Mnangagwa told public television outside the polling booth.
Mugabe was forced to resign when ruling ZANU-PF lawmaker threatened to impeach him after military generals launched a brief takeover to install their favoured candidate, Emmerson Mnangagwa.
Many in Zimbabwe knew no other leader but Mr Mugabe, who led the country for 37 years and since independence from white minority rule in 1980. "President! The president is here!"
"I am very happy that the process for campaigning was peaceful (and) voting today is peaceful", he added. "We are disturbed by false reports alleging that the Zimbabwe Defence Forces is going to be used by ZANU-PF to rig the posted vote". "We have suffered enough", he says as as he walks off on to the winding dirt road.
He swore he would not support the ruling party he controlled for decades at the polls, saying: "I can not vote for 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.
The European Union, the United States and the Commonwealth have sent observers. I have to see a better Zimbabwe for my kids.