IRI-NDI Zimbabwe Election Observer Rep. Bass Cited by The Wall Street Journal

“I’m really happy,” she said outside her polling station in a rural township in eastern Zimbabwe, showing off the dark-blue circle election officials had inked on her left pinkie. “I feel like I’m really free to vote.”

Across Zimbabwe, millions of voters marked their ballot papers in their first election without ousted strongman Robert Mugabe’s name at the top—and, many said, without the widespread intimidation that had marred previous polls.

Now, voters said, they are counting on the electoral commission, and international observers, to ensure that their ballots are counted without manipulation by the powerful ruling party.

“That’s the big question,” said Emmanuel Chidamahiya, 57, who had plunked himself down on a fold-up stool, three hours into waiting in line to vote on the outskirts of the capital, Harare. “They have to prove themselves.”

Results are expected later this week. The electoral commission said that one hour before polling stations closed, voter turnout averaged 75%.

President Emmerson Mnangagwa, a former Mugabe ally who was installed by the military in November, was one of two leading candidates in an early-July poll, 3 percentage points ahead of Nelson Chamisa, a lawyer and part-time pastor who leads the Movement for Democratic Change, or MDC.

Holding a free and fair vote has been one of the central campaign pledges of Mr. Mnangagwa and his ZANU-PF party, but opposition parties and civil-society groups have questioned whether that commitment will hold if that means handing over power to his opposition rival.

Zimbabwean election watchdogs reported isolated incidents of intimidation, including voters being forced to seek assistance in completing their ballots, and some polling stations having been moved to a different location. But they said voting went relatively smoothly.

The election “appears to be one of the most credible electoral processes we have seen for over a decade in Zimbabwe,” said Nicole Beardsworth, a postdoctoral researcher at the University of York in the U.K. who was in Harare for the vote. “However, it is important to note that it is at the tallying, collating and reporting stage of the vote that things can still go drastically wrong.”

ZANU-PF’s 38 years in power have left the economy in tatters. Unions say at least eight out of 10 Zimbabweans are unemployed. The public-sector payroll eats up as much as 90% of government revenue and banks have curtailed withdrawals of U.S. dollars, the dominant currency since the Zimbabwean dollar was abolished in 2009 after years of hyperinflation.

Mr. Mnangagwa, 75 years old, argued during his campaign that his decades in government give him the experience to secure foreign support, including a bailout from the International Monetary Fund, and to bring in investors. Polls suggest he remains popular with older voters and his ZANU-PF maintained a significant lead in rural areas.

“I voted for Mnangagwa. I’m giving him a chance,” said a 59-year-old man who declined to give his name, after casting his ballot in Harare. “We want to have a country where the guys who are leaving university get jobs.”

The 40-year-old Mr. Chamisa, meanwhile, cast himself as a youthful change maker. His campaign promises, including bringing bullet trains to a country whose roads are pockmarked with potholes, won over many young and urban voters.

“Zimbabwe has suffered for about 38 years,” said Divine Charwadza, a 25-year-old banking and finance graduate who supported Mr. Chamisa. “It’s time for something new.”

Many Zimbabweans worry that the military might intervene to stop Mr. Chamisa from taking power. In a disputed 2008 election, human-rights groups say, more than 150 opposition supporters were killed after Mr. Mugabe was forced into a runoff with the MDC. Others are anxious that disgruntled opposition supporters could riot if they feel cheated out of their vote.

If Mr. Mnangagwa wins, “we must go to the street and remove the president,” said Joseph Ingirande, a 26-year-old MDC supporter.

Some local democracy activists have said they worry that international observers, including from the U.S. and the European Union, might be reluctant to give a thumbs down for the electoral process to avoid destabilizing the mineral-rich nation.

U.S. Congresswoman Karen Bass, who is part of the observer mission from the National Democratic Institute and the International Republican Institute, said election monitors wouldn’t ignore meaningful irregularities.

“To come all this way to then sugarcoat something that doesn’t help the process or stability,” said Ms. Bass. “I don’t think you can have stability without free and fair elections.”