Mr Hollande - who got an estimated 52% of votes in Sunday's run-off - said the French had chosen "change".
Admitting defeat, centre-right incumbent Nicolas Sarkozy wished "good luck" to Mr Hollande.
Analysts say the vote has wide implications for the whole eurozone. Mr Hollande has vowed to rework a deal on government debt in member countries.
Shortly after polls closed at 20:00 (18:00), French media published projections based on partial results giving Mr Hollande a lead of almost four points.
Exuberant Hollande supporters have gathered on Place de la Bastille in Paris - a traditional rallying point of the Left - to celebrate.
Mr Hollande capitalised on France's economic woes and President Sarkozy's unpopularity.
The socialist candidate has promised to raise taxes on big corporations and people earning more than 1m euros a year.
He wants to raise the minimum wage, hire 60,000 more teachers and lower the retirement age from 62 to 60 for some workers.
Mr Hollande has also called for a renegotiation of a hard-won European treaty on budget discipline championed by German Chancellor Angela Merkel and Mr Sarkozy.
In his concession speech, Mr Sarkozy told supporters: "Francois Hollande is the president of France and he must be respected."
The outgoing president said he was "taking responsibility for defeat".
Hinting about his future, he said: "My place will no longer be the same. My involvement in the life of my country will now be different."
During the campaign, he had said he would leave politics if he lost the election.
Mr Sarkozy, who has been in office since 2007, had promised to reduce France's large budget deficit through spending cuts.
It is only the second time an incumbent president has failed to win re-election since the start of France's Fifth Republic in 1958.
The last was Valery Giscard d'Estaing, who lost to socialist Francois Mitterrand in 1981. Mr Mitterrand had two terms in office until 1995.
The new president is expected to be inaugurated later this month.
A parliamentary election is due in June.
Source: BBC News
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