Francois Hollande got 28.4% of votes, according to projections based on partial results, against 25.5% for centre-right incumbent Nicolas Sarkozy.
Nicolas Sarkozy and Francois Hollande will face each other in a second-round run-off on 6 May.
The election has been dominated by widespread anxiety over the French economy and the wider eurozone crisis.
The estimates – based on votes counted in polling stations that closed early at 18:00 – were announced by French media when all voting ended at 20:00.
It is the first time a French president running for re-election has failed to win the first round since the start of the Fifth Republic in 1958.
Nicolas Sarkozy – who has been in power since 2007 – was facing a total of nine candidates in Sunday’s first round.
Far-right leader Marine Le Pen came third with about 20% of the vote, and leftist candidate Jean-Luc Melanchon fourth with more than 11%.
Centrist Francois Bayrou, who was hoping to repeat his high 2007 score of 18%, garnered only about 9%.
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