She is the front-runner among candidates for the presidential election in 2017 but opinion polls suggest she would lose a run-off vote.
Alarm bells - BBC Europe editor, Katya Adler
The EU worries Brexit could reverse 70 years of European integration.
In all my years watching European politics, I have never seen such a widespread sense of Euroscepticism.
Plenty of Europeans looked on with envy as Britain cast its In/Out vote. Many of the complaints about the EU raised by the Leave campaign resonated with voters across the continent.
Across Europe leading Eurosceptic politicians queued up this morning to crow about the UK referendum result.
But the mood in Brussels is deeply gloomy. The Brexit vote sends screaming alarm bells, warning that the EU in its current form isn't working.
Last Friday, Ms Le Pen had told a gathering of far-right parties in Vienna: "France has possibly 1,000 more reasons to want to leave the EU than the English."
She said the EU was responsible for high unemployment and failing to keep out "smugglers, terrorists and economic migrants".
Mr Wilders, leader of the Party for Freedom in the Netherlands, said in a statement: "We want to be in charge of our own country, our own money, our own borders, and our own immigration policy.
"As quickly as possible the Dutch need to get the opportunity to have their say about Dutch membership of the European Union."
The Netherlands faces a general election in March and some opinion polls suggest Mr Wilders is leading. A recent Dutch survey suggested 54% of the people wanted a referendum.
Mateo Salvini, the leader of Italy's anti-immigration Northern League, tweeted: "Hurrah for the courage of free citizens! Heart, brain and pride defeated lies, threats and blackmail.
"THANK YOU UK, now it's our turn."
The populist Danish People's Party, which wants renegotiation with the EU, hailed a "courageous" decision by the British, but also said that everyone should "keep their heads".
Reacting to the vote, UK Independence Party leader Nigel Farage said "the EU is dying".
But Donald Tusk, the president of the European Council, said this was "not a moment for hysterical reactions".
He said he was "determined to keep our unity as 27" members, adding that: "Until the UK formally leaves the EU, EU law will continue to apply to and within the UK, and by this I mean rights as well as obligations."
European Parliament President Martin Schulz denied Brexit would trigger a domino effect, saying the EU was "well-prepared".
But Beatrix von Storch, of Germany's Eurosceptic AfD party, praising "Independence Day for Great Britain", demanded that Mr Schulz and European Commission head Jean-Claude Juncker resign.
"The European Union has failed as a political union," she said.