Preliminary results pointed to the measure being approved, in what would be a defeat for the far right nationalist Swiss People's Party (SVP), which put issues of Islam and national identity at the centre of the debate.
The government as well as most lawmakers and political parties supported the proposal that would allow the grandchildren of immigrants to skip several steps in the lengthy process of securing a Swiss passport.
According to a migration department study, less than 25,000 people in the country of about eight million currently qualify as third generation immigrants, meaning they have at least one grandparent who was born here or acquired Swiss residency.
Nearly 60 percent of that group are Italians, followed by those with origins in the Balkans and Turkish nationals.
Debate on the proposal had nothing to do with religion at the outset, said Sophie Guignard of the Institute of Political Science at the University of Bern.
It was the SVP, a party repeatedly accused of demonising Islam, that focused on the risks of more Muslims becoming citizens and the possible "loss of Swiss values", Guignard told AFP.
Polls closed at midday and most in the wealthy Alpine nation had already voted by mail.
The gfs.bern polling institute reported that the early trend indicated a win for the "Yes" camp.
Eight cantons including major population centres like Geneva, Zurich and Basel voted to approve the measure with two small cantons voting "No", final results showed.
A change to citizenship laws requires a constitutional amendment, meaning the Yes side needs to win both a majority of votes and a majority of Switzerland's 26 cantons.
Sunday's referendum is one of four each year for voting on subjects affecting federal as well as local laws and institutions.
The No camp faced heavy criticism over a widely-distributed poster showing a woman staring out from a black niqab with a tagline urging voters to reject "uncontrolled citizenship".
The SVP is not officially responsible for the poster.
It was commissioned by the Committee Against Facilitated Citizenship, which has several SVP members including in leadership positions.
The co-chair of that committee and an SVP lawmaker, Jean-Luc Addor, urged people to vote against the measure on grounds that one day most third generation immigrants will not be of European origin.
"In one or two generations, who will these third generation foreigners be?" he wrote in an opinion piece on the SVP website.
"They will be born of the Arab Spring, they will be from sub-Saharan Africa, the Horn of Africa, Syria or Afghanistan," said Addor, who has defended the niqab poster.
Guignard said mainstream politicians and journalists view the poster as "a violent attack against Muslims".
Political initiatives that either directly or implicitly target Muslims may be on the rise in the West, notably including US President Donald Trump's travel ban against seven mainly Muslim countries, which was undone in court this week.
But in Switzerland such moves are nothing new.
The SVP in 2009 successfully persuaded Swiss voters to approve a ban on new mosque minaret construction, while religiously-charged messages have been a part of multiple referendums on immigration since.
Polls before the vote indicated a win for proponents of the citizenship measure but the No camp had gained ground in recent weeks.