The former president was lying third in the US-style primary after more than 80% of the votes had been counted.
Mr Sarkozy, 61, told his supporters: "I failed to convince a majority of voters."
It leaves two men who will go forward to a second round - one of whom is almost certain to be one of the two contenders for the next president.
They are Francois Fillon, who was leading the count with 44.1% of public votes, and Alain Juppe, who has so far secured 28.2% of the votes.
Mr Fillon, 62, served as prime minister under then president Sarkozy from 2007 to 2012, while Mr Juppe, 72, was prime minister from 1995 to 1997 under president Jacques Chirac.
Unless Mr Fillon ends up with more than 50%, it is expected that the pair will now face each other again on 27 November.
Whoever wins that vote is most likely to go head-to-head with far-right leader Marine Le Pen, who is predicted to be one of two in the final round of the French presidential election, next May.
The ruling Socialist party, under the current President Francois Hollande, is in disarray, with polls showing he is likely to be eliminated in early voting next year.
Ms Le Pen is campaigning on a pro-nationalist, anti-immigration ticket amid growing fear of Islamist terrorism.
Her rise has been attributed to the same sentiment that prompted Donald Trump's victory in the US and Brexit in the UK.
Last week she told Sky News that Brexit and Mr Trump's shock victory were signs of the "emergence of a new world" in which disillusionment with mainstream politics has led many in France to consider her their only hope.
For months, pollsters have been predicting Mr Juppe would win the primaries and subsequently defeat Ms Le Pen.
His second place will further cast doubt over whether the polls are able to predict the outcome - something they failed to do in the UK's EU referendum and in the US election.
Several analysts say that Mr Juppe is the most likely to be able to beat Ms Le Pen, if the pair were to face each other.
One analyst, Charles Lichfield of Eurasia Group, gives Ms Le Pen a 25% likelihood of beating Mr Juppe. But, he said, if she goes up against Mr Fillon, her chances of winning jump to 35%.
Marine Le Pen becoming President would badly damage Europe's economy, Austria's Social Democrat Chancellor Christian Kern said earlier on Sunday.
Others have suggested it could even lead to the break up of the eurozone.
Voter turnout in what was France's first ever conservative presidential primary, was high on Sunday, topping 2.5 million by 5pm.
Other candidates in the vote were Nathalie Kosciusko-Morizet - the only woman on the conservative ballot; former government ministers Bruno Le Maire and Jean-Francois Cope, and Parliament member Jean-Frederic Poisson.
Mr Sarkozy and Mr Le Maire, after both conceding defeat, threw their weight behind Mr Fillon. Ms Kosciusko-Morizet, a former environment minister, endorsed Mr Juppe.
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