The "yes" vote is widely expected to win, with a poll on Friday showing as many as 90 percent of people in favour.
The Social Democratic Party (PSD) has opted to let voters also cast their ballots on Sunday to ensure maximum turnout, as at least 30 percent is required for the result to be valid.
Turnout at 4 pm (1400 GMT) on Saturday, however, was under four percent, according to official figures. There are an estimated 19 million voters.
Among the first to cast their votes was PSD strongman Liviu Dragnea.
"The time has come to decide ourselves how we want to live in our country," he said, adding that a "yes" vote was "absolutely not a vote against a minority".
From a legal point of view, nothing will change if the "yes" side wins the referendum.
Same-sex couples are already not allowed by law to marry or enter into civil partnerships in Romania.
Nevertheless, critics say a change in the wording of the constitution will make it difficult or nigh-on impossible for gays and lesbians to marry in future.
The country's LGBT community, which already complains that gay people are subject to widespread discrimination on an everyday basis, believes the referendum -- which has the explicit backing of the Orthodox church -- will fuel homophobia still further.
Romania, which joined the European Union in 2007 and is the bloc's second-poorest member after Bulgaria, only decriminalised homosexuality in 2001.
A defeat would deal a severe blow to the PSD who have been campaigning, albeit unofficially, alongside Orthodox priests for the "yes" side.
"My Orthodox education and my traditional upbringing make me say 'yes'," Dragnea said earlier.
It was Dragnea, 55, who led the PSD to a sweeping victory in 2016 elections.
But he was unable to run for the post of prime minister due to legal troubles, including a two-year suspended prison sentence for vote-rigging in a referendum in 2016.
And he is scheduled to appear in court on Monday -- the same day the result of the referendum is expected to be announced -- to appeal another sentence, of three-and-a-half years, over a fake jobs scandal.
The government's decision to press ahead with the referendum has alarmed Brussels, with the EU Commission's deputy chief, Frans Timmermans, reminding Bucharest of its human rights commitments.
"I don't want family values to be transformed into arguments that encourage the darkest demons and hatred against sexual minorities," he said during a debate on a series of reforms that are seen as undermining the independence of Romania's judiciary.
Sociologist Marius Pieleanu of the Avangarde institute warned that such interjections could actually have the reverse effect as eurosceptic sentiment gains ground in the country.
"Some undecided voters might swing to the 'yes' side precisely because they feel such attitudes are hostile towards Romania," he told AFP.
If the anti-gay marriage lobby has been ramping up its rhetoric in the run-up to the vote, civil rights groups have urged a boycott.
"In a democracy, the rights of minorities are not put to a vote. That's the difference between the Middle Ages and the 21st century," said the Centre for Legal Resources, a non-profit NGO.
jpegMpeg4-1280x720Bela Marko, a poet and former president of the Ethnic Hungarians' Union in Romania, warned that "everything will change the day after the vote, as other initiatives will follow, first against abortion, then on the state's religion, the death penalty, the Roma" and other issues.