Bronislaw Komorowski, who has been acting leader since President Lech Kaczynski died in a plane crash on 10 March, won but by less than expected.
With 94% of votes counted, he had won 41.22%, compared to the 36.74% taken by Kaczynski's twin brother, Jaroslaw.
The two will now go head-to-head in a run-off on 4 July.
Lech Kaczynski was one of 96 people who died in a plane crash in Smolensk in western Russia on 10 April.
The dead included many of Poland's political and military leaders, and the disaster has overshadowed the election campaign and made it one of the country's least aggressive, BBC Warsaw correspondent Adam Easton said.
'National unity'
The disaster has also boosted the ratings of the late president's twin, Jaroslaw, a combative conservative former prime minister who was voted out of office three years ago.
His personal loss has changed him, our correspondent says, and he is presenting himself as a man of compromise.
Mr Kaczynski voted on Sunday in Warsaw accompanied by his late brother's daughter and two granddaughters.
But the front-runner was always the government's candidate, Bronislaw Komorowski, who is a member of the ruling Civic Platform party.
The centre-right parliamentary speaker, who is now the acting president, says his whole career proves that he is a man who can unite the nation.
Both candidates appealed to national unity during an election overshadowed by April's plane crash and recent floods that killed 24 people and forced thousands from their homes.
Turnout in the election was reported to be just short of 55%. More than 30 million of Poland's 38 million citizens were registered to vote.
