The alliance took 32.5 percent of the vote according to the latest polls, surging ahead after pre election polls put it neck and neck with the anti-EU and anti-immigration National Front (FN) led by Marine Le Pen.
The latest figures suggest the FN won about 25.35 percent, behind the 30 percent or more it was tipped to win from French voters who are dissatisfied with the stagnant economy and issues surrounding immigration.
The far-right party led the first-round vote in 43 out of 98 "departments", which have power over local issues such as school and welfare budgets, according to interior ministry figures.
The FN is expected to go through to the second round in more than half of the 1,100 "cantons", an administrative division below "departments", that will vote again in a week.
The party's best results "are concentrated in the southeast, particularly in the cities and near the cities it runs," said political scientist Jean-Yves Camus.
Provisional interior ministry figures gave the Socialists and their left-wing allies around 22 percent of the vote.
The mainstream parties will be able to call on smaller allies when voters return for run-off elections next Sunday, while the FN will struggle to find partners.
In an address last night, UMP leader Nicolas Sarkozy again clearly ruled out any electoral pacts with the extreme right, declaring: "There will be no local or national deal with the leaders of the FN."
Although the FN won even more votes than in European elections last May, on a slightly higher turnout, there was some relief among the mainstream right and left parties that the FN score was not as high as some surveys had predicted.
"Tonight, the far-right, even if it is too high, is not the leading political party in France," said Socialist Prime Minister Manuel Valls.
He called on voters to back either the left or right in next week's second-round run-offs to keep the far right from power.
FN leader Marine Le Pen remained bullish about the initial results, pointing to the fact they were higher than the party's victorious tally in last year's European polls.
"This massive vote for the National Front that is taking root in election after election shows that the French want to rediscover their freedom," she said.
Her party has capitalised on anger over France's lacklustre economy, as well as the politically explosive issues of immigration and the integration of Islam into French society after the Paris terrorist attacks.
But it has also benefited from Hollande's disastrous popularity figures.
His ratings have hit record lows, despite a temporary boost in the wake of the January jihadist attacks on the Charlie Hebdo magazine and a Jewish supermarket, when he was credited with rallying the country.