"This evening's result confirms the success of the left," Francois Fillon, the French prime minister, said in a televised address.
"That is a disappointment. I assume my share of responsibility."
Sarkozy, whose UMP party still has a comfortable majority in the national parliament, has insisted that the regional poll is not a verdict on central government.
However, he is expected to order a reshuffle of his cabinet in the next few days.
Fears about job losses, immigration and security, as well as resentment about issues such as bank bailouts and executive pay have all dented Sarkozy's popularity, which has fallen since he won approval for his handling of the financial crisis in 2008.
Martine Aubry, the leader of the Socialists, said that the result showed that many of his policies must now be rethought.
"The French people have spoken. They must be heard," she said.
"Listening to what the French people have to say means a profound change in policies."
But Fillon said employment and growth would be the "absolute priority" for the government and pledged to push ahead with the changes to pensions and social welfare promised by the president when he came to power in 2007.
"You don't govern a big country like France according to regional elections, but by sticking to the course fixed by national elections," he said.
The powerful CGT union has called for a day of protest over wages and pensions on Tuesday.
Other groups, such as farmers, may also see a big defeat for the centre-right as an opportunity to put pressure on the government.
|French voters appear frustrated with Sarkozy's handling of the economic crisis [AFP]
There has also been increasingly open criticism of the leadership from within the UMP, where some well-known party figures including Alain Juppe, the former prime minister, are calling for a change of course.
Jean-Francois Cope, the head of the UMP parliamentary group and one of Sarkozy's main rivals in the party, said: "We can't hide from this.
"This is obviously a real defeat for us. You can't beat about the bush. That's the reality of it."
The centre-right suffered one of its worst losses in years in the first round of the ballot on March 14, with Sarkozy's party scoring just 26 per cent as the Socialists took 29 per cent and the broader left parties combined claimed some 50 per cent.
Regional councils in 25 regions, 22 on the French mainland and three overseas territories, were up for grabs, the Caribbean island of Guadeloupe having already been won outright by the Socialists in the first round of voting.