With 88 per cent of the ballots counted, Rousseff had 40.7 per cent of the vote and Neves 34.7 per cent, said the Supreme Electoral Tribunal (TSE).
That left Silva, a one-time maid whose meteoric rise looked set to make her multi-racial Brazil's first black president just one month ago, on track to be eliminated from the race with 21 per cent.
Neves, the scion of a political dynasty and business-world favorite, meanwhile survived a turbulent campaign that nearly relegated him to the dustbin of also-rans to reach a likely October 26 run-off against the incumbent, whose management of the world's seventh-largest economy he has sharply criticised.
The race for the second spot in the run-off went down to the wire, with Neves, a popular ex-governor and handsome husband of a former model, battling back from third place to keep his presidential hopes alive.
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Silva meanwhile suffered a spectacular setback from her whirlwind late entry into the race.
Silva, a one-time maid who vowed to be Brazil's first "poor, black president," upended the race when she replaced the Socialist Party's original candidate, Eduardo Campos, after he was killed in a plane crash on August 13.
But Rousseff and Neves steadily reversed her lead, with the latter pulling ahead of her for the first time in the final three polls, released Saturday.
The same polls found Rousseff would defeat Neves in a run-off by more than five percentage points.
The election, the closest in a generation for Latin America's largest democracy, is widely seen as a referendum on 12 years of Rousseff's Workers' Party (PT).
The party endeared itself to the masses with landmark social programs and an economic boom in the 2000s that have lifted 40 million Brazilians from poverty, increased wages and brought unemployment to a near-record low.
Rousseff, a former guerrilla who was jailed and tortured for fighting the country's 1964-1985 dictatorship, has also been battered in recent weeks by a corruption scandal implicating dozens of politicians - mainly her allies - at state-owned oil giant Petrobras.