But a television exit poll suggested that the country could still be on its way to rejecting membership of the euro, despite the killing of the foreign minister who tirelessly campaigned for a yes vote.
The poll by Swedish public television, which questioned 7,000 voters as they left polling stations, gave the 'no' camp 51.8% and the 'yes' camp 46.2%, with 2% handing in blank votes.
Deputy Finance Minister Gunnan Lund, a leading campaigner for the yes vote, however said the exit poll results should be treated cautiously.
"We have seen thse kind of polls show crazy results in the past," he said.
An upset "Yes" vote in the broadly EU-sceptic Nordic nation would make it the first European Union state to endorse the 12-nation euro at the ballot box since its launch in 1999. A "No" would keep it outside with Britain and Denmark.
The Danes rejected the euro in a September 2000 referendum, which the "No" side won by 53% to 47%.
Killer at large
Sweden had looked likely to return a resounding "No" until Lindh, the popular 46-year-old foreign minister and mother of two, was knifed to death while shopping at a Stockholm department store.
Police hunting her killer circulated security video pictures of a dark-haired man in a baseball cap and grey college sweater.
The man remained at large on Sunday, evoking painful memories of the 1986 assassination of Prime Minister Olof Palme, a few blocks away by gunshot. Palme's killer has not been found.
Politicians suspended all campaigning after Lindh died early on Thursday, and a wave of emotion swept the nation.
Flowers, notes and children's drawings piled up outside the store in central Stockholm, and teary-eyed Swedes paid their respects to the woman who had been tipped as the next prime minister.
Some opinion polls showed a surge in support for the "Yes" side, which has lagged the "No" side since April.
Police have released images of a suspect in the murder of Lindh
On Saturday, a Gallup poll gave the "Yes" side, led by Prime Minister Goran Persson and supported by the political establishment and big business, a 43%-42% lead.
But a Temo poll saw the "No" camp, with the support of the left, Greens and many women, winning by 46%-40%.
"I'm voting 'No'," said Birgitta Henriksson, 50, shopping at a market early on Sunday. "I think the murder will make many undecided people vote 'Yes'. I liked her too but that's no reason to change my decision on the euro."
Until Lindh's murder, debate had focused on whether the euro would help or hinder Sweden's relative economic advantages over the euro zone - which has higher unemployment and lower growth - and what it would mean for Sweden's cherished cradle-to-grave welfare state, funded by some of the world's highest taxes.
Eurosceptics say handing over monetary tools to the European Central Bank would leave them unshielded from economic shocks, while pro-euro Swedes think joining the final phase of Economic and Monetary Union would boost trade and ensure future growth.
"I think the 'No' side will win but it will be very close because of the murder of Anna Lindh," said Bengt Klingvall, a 42-year-old social worker who voted "No" by postal ballot. "We have a strong economy, we should stay free of the euro area."
About seven million people are eligible to vote. Turnout in Swedish referendums is traditionally around 80%.
All parties have vowed to respect the outcome, which in the case of a "Yes" result would mean Sweden switching to euros for all transactions in 2006 at the earliest.