Eurosceptic parties also made big gains in other European countries – also coming first in Denmark.
The centre-right EPP looked set to be the biggest bloc in parliament.
Turnout in the election was 43.1%, according to provisional European Parliament figures – up on last time.
That would be the first time turnout had not fallen since the previous election – but would only be an improvement of 0.1%.
In France, the National Front (FN) leader Marine Le Pen said after seeing exit polls: “Clearly we are in the lead.”
A statement by her party accused the French government of “massive fraud” and “industrial scale” vote-rigging, saying PM Manuel Valls had tried to prevent the National Front winning “by the most odious means”.
It said in many polling stations voters were given incorrect papers or did not get ballots that included the National Front.
The election is the biggest exercise in multi-national democracy in the world.
The 751 seats are allocated in proportion to each country’s population.
The vote will affect the lives of the EU’s 500 million citizens, and the chamber has much more power than it used to.
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