Gianni Infantino caused a surprise by polling the most votes in round one of voting in Friday's election to succeed Sepp Blatter as Fifa president.
The Uefa secretary general failed to get enough to seal outright victory, claiming 88 of the 207 votes available at an extraordinary congress in Zurich.
Pre-vote favourite Sheikh Salman bin Ebrahim al-Khalifa won 85.
Prince Ali bin al-Hussein was next with 27, followed by Jerome Champagne on seven. Tokyo Sexwale withdrew earlier.
It is the first time voting for the presidential election had reached a second round since 1974, when Joao Havelange of Brazil became the first non-European president ahead of England's Sir Stanley Rous.
Blatter, who led world football's governing body since 1998, stood down last year and was later banned from football for six years.
"Infantino's camp has consistently maintained an air of optimism throughout the final days of this election campaign," said BBC Radio 5 live's sports news correspondent Richard Conway.
"They were not surprised at how well he performed in round one.
"What was surprising according to those close to Shaikh Salman is that pledges of support from Asia and Africa failed to materialise for the Bahraini."
What happens now?
To become president after the first round of voting, a candidate needed to secure two-thirds of the available votes, which equated to 138.
In round two, a simple majority is required, which means Infantino needs another 16 votes to become Fifa's ninth president.
"This election is being fought on the floor of the congress hall right now, with supporters of both Shaikh Salman, Gianni Infantino and Prince Ali talking to voterw," added Conway. "We could be in for a long night."
What else has happened?
Before voting began, reforms were passed to help make Fifa a more transparent and accountable organisation.
All salaries will be disclosed, while a limit of four years has been placed on a president's term.
A new council to replace the current executive committee has also been introduced, featuring a female representative from each confederation.
Greg Dyke, who will cast a vote as chairman of the English Football Association, says the reforms are "more important" than the new leader as it will provide an "opportunity for Fifa to start again".
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How does voting work?
There are 209 Fifa nations but Kuwait and Indonesia are currently barred from taking part, meaning there are 207 eligible voters.
To become president after the first round of voting, a candidate must secure two-thirds of the available votes. If no candidate achieves that mark, a simple majority is required in the second round.
If there is still no winner, a third round will take place, minus the candidate with the fewest votes in round two.
Fifa says a winner must be declared on Friday because an ice hockey rink is due to be installed at the Hallenstadion venue at midnight.
How significant is Friday's election?
Given everything that has happened to Fifa, this is seen as a pivotal moment for an organisation which has been heavily criticised for its lack of transparency and for failing to clamp down on corruption within it.
A new leader, together with a raft of reform measures, is seen as a chance to start afresh.
Acting president Issa Hayatou said Friday can "signal a new dawn", adding: "This is our opportunity to show we are united in building a stronger Fifa."
How bad has it got for Fifa?
There have been widespread allegations of corruption, the arrest of leading officials, the banning of its president and the sight of big-name sponsors deserting the organisation.
Numerous Fifa officials have been indicted in the United States, while Swiss authorities are also investigating the organisation.
Blatter has also been banned from all football activity for six years after being found guilty of breaching Fifa's ethics rules over a $2m (£1.3m) "disloyal payment" to the head of European football's governing body Uefa, Michel Platini, who had been favourite to succeed the Swiss.
Former France captain Platini was also suspended. Both men deny any wrongdoing and are appealing to the Court of Arbitration for Sport.
Furthermore, Jerome Valcke, Fifa's secretary general and formerly Blatter's right-hand man, was banned for 12 years following allegations - which he denies - of misconduct while in office.
How important is the president?
Fifa's leader is the figurehead for world football, often seen in public alongside presidents, prime ministers and royalty.
There have been eight of them so far, presiding over the organisation's executive committee, which is where the real decision-making power lies.
Fifa organises World Cups and other international tournaments, distributes broadcasting rights and should both protect and develop the world's most popular sport.
The president also "legally represents" the organisation, "maintains relations between Fifa and the confederations, members, political bodies and international organisations", and "implements the decisions passed by the congress and the executive committee".
Who wants to take over?
There are four candidates following Sexwale's withdrawal. They are:
- Prince Ali bin al-Hussein: Aged 40, president of the Jordanian Football Association
- Jerome Champagne: Aged 57, a former Fifa executive from France
- Gianni Infantino: Aged 45, the Swiss is Uefa's general secretary
- Sheikh Salman bin Ebrahim al-Khalifa: Aged 50, the Bahraini is Asian Football Confederation president
What do they say?
Prince Ali: "I'm a candidate beholden to no one. I wouldn't apply political pressure or coercion. I'm the only candidate from a national association."
Infantino: "I'm not a politician, I'm football person and I'm a worker. If we stop doing politics and start doing football, the world will admire us."
Sheikh Salman: "My past and my track record speaks for itself. We want someone who is responsible and can deliver the promises he says."
Champagne: "I want a Fifa that serves football, that serves you. The Fifa I dream of is one which correct the inequalities."
What do they stand for?
Prince Ali wants to quadruple the amount Fifa's member associations receive - believing it will increase their sustainability - but wants the money properly accounted for.
Infantino would expand the World Cup to 40 teams to ensure more smaller nations can participate. He also wants to hugely expand Fifa's development plan by investing £860m of its revenues and giving £3.6m to each member association.
Sheikh Salman's big idea is to split Fifa in two with a business side handling commercial issues and the football side organising World Cups and developing the game. He believes this would stop executives making self-interested decisions.
Champagne's most recent manifesto emphasised "rebalancing" the inequality in football and "reconciling" the game's "protagonists". He wants to introduce technology to help referees and appoint women to key Fifa roles. He also wants Fifa to be run like a public sector organisation.
Will it really be a new era?
BBC sports editor Dan Roan in Zurich:
"Few fans or players would recognise these individuals, let alone know about their policies or have trust in their leadership.
"The selection of Sepp Blatter's successor should be the moment the governing body finally consigns the tainted tenure of their former president to history and symbolically moves on from the stranglehold he held over the organisation for so long. Except to many, it simply does not feel like that.
"All of the men running for president are members of the football establishment. Four of them have spoken to Blatter in the build up to the election, presumably to ask for advice. All are loathe to condemn the disgraced former president.
"The next 24 hours matters a great deal, and much is at stake. But do not assume that it represents the end of this great scandal, or the solution to FIFA's troubles. We should all know better by now."