The conservative UMP was credited with around a fifth of the vote, while President Francois Hollande's Socialists trailed an abysmal third, with around 14 per cent.
"Our people demand only one sort of politics - the politics of the French people, for the French people and with the French people," said Marine Le Pen, daughter of the party's firebrand founder Jean-Marie Le Pen.
"They have declared they no longer want to be ruled from the outside, to have to bow to laws they did not vote for, or to kowtow to [EU] commissioners who are not subject to the legitimacy of universal suffrage."
In Britain, which voted on Friday, the UK Independence Party (UKIP) was ahead of the governing Conservatives and opposition Labour Party.
UKIP leader Nigel Farage described the breakthrough as an "earthquake". He said: "Never before in the history of British politics has a party seen as an insurgent party ever topped the polls to a national election".
In Denmark, the anti-immigrant Danish People's Party won around 23 per cent of the vote, enough to gain three of the country's 13 seats, according to pollsters. In Germany, Alternative for Germany, a small party demanding a return to the deutschmark and the end to German bailouts for debt-stricken EU countries, scored above the 5 per cent threshold needed to gain seats. But Chancellor Angela Merkel's conservatives were comfortable winners, and in Italy, the anti-establishment Five Star Movement did well but was outpaced by Prime Minister Matteo Renzi's centre-left Democratic Party.
The Parliament will remain in the control of the two mainstream centre-right and centre-left parties that are solidly behind European integration and have co-dominated the assembly for decades. Estimates gave the two blocs 212 and 186 seats respectively.
But the hefty protest vote will be seen not only as a sign of anger towards unpopular national governments but also as hostility towards Brussels, battling a reputation for being smug, bureaucratic or dictatorial.
Many countries remain in the grip of an economic slump. The EU counts 26 million unemployed, and in some regions more than one in two of the under-25 population of working age is without work. Even in Greece, which is surviving thanks to EU bailouts, the big gainers yesterday were the anti-austerity left-wing Syriza Party and the neo-Nazi Golden Dawn.
"The relatively strong showing of nationalists is a disturbing sign of many people feeling disenfranchised," Erik Nielsen of Italy's UniCredit Bank said in an analysis.
The eurosceptics' success seems likely to plunge the Parliament into a period of turbulence, where the EU's core principles will come under raucous challenge.
But, said Nielsen, the pro-EU majority means the Parliament's fundamental workings are unlikely to be affected.
Invisible to millions of citizens, the European Parliament has evolved under successive treaties from being a consultative assembly to a directly-elected legislature with powers of "co-decision" in 80 per cent of EU-wide laws and measures.
But it still plays second fiddle to the Council of Ministers, which gathers the governments of the 28 member-states and is answerable to home parliaments.
The Parliament did its best to stir interest in its new right to propose candidates for president of the European Commission, the EU's powerful executive arm. But turnout was a wretched 43.11 per cent, only a tiny gain on the record low of 43 per cent in 2009.
Tomorrow, EU heads of state will discuss the next commission chief and allotting portfolios among the 28 commissioners. They will also discuss who will succeed Herman van Rompuy, president of the council, and Catherine Ashton, the EU foreign policy chief, both of whom are widely viewed as ineffectual.
The European elections
• Mainstream centre-right bloc forecast to win 212 of 751 seats, down from 274, centre-left 186.
• Mainstream parties won enough seats to muster a majority on issues on which they agree.
• Eurosceptic parties' support up on last elections in 2009.
• 43.11 per cent turnout.
France
National Front was the outright winner with 26 per cent, nearly 6 points ahead of the UMP conservative party of former President Nicolas Sarkozy.
Britain
Nigel Farage's UKIP topped the poll with 30 per cent, 12 per cent higher than in 2009. The Liberal Democrats suffered big losses, trailing behind the Greens. The BNP's Nick Griffin lost his seat.
Denmark
The Social Democrats retained their five seats to remain the biggest party. The populist, opposition Danish People's Party won three more seats for a total of four.
Germany
A year-old party that wants that country to stop using the euro won 6.7 per cent of the vote.
Greece
The leftist Eurosceptic Syriza party led with 26.49 per cent. The extreme right Golden Dawn was third with 9.33 per cent.
The Netherlands Geert Wilders' right-wing Eurosceptic Party for Freedom dropped a seat from five to four.
Italy
The Democrats appear to have beaten off a challenge by the anti-euro 5-Star Movement of comic Beppe Grillo. The Democrats were forecast to win 40 per cent.
Spain
The leftist United Left and We Can parties took 11 of the country's 54 seats. The Popular Party took 16 and the Socialists won 14.
- AP
