The leaked documents, called the 'Panama Papers', suggest that thousands of wealthy people have established companies in tax havens, including Panama, with a view to either avoiding or evading tax in their home country.
MEPs immediately urged the European Commission and the EU to adopt "ambitious" rules to improve corporate transparency and the fight against tax avoidance by multinationals.
EU transparency campaigners say the probe has shown how a network of lawyers, bankers and other facilitators around the world "help the corrupt to hide illicit wealth."
They say it highlights the urgent need to "open up" corporate ownership to the public, in line with recent EU legislation.
The revelations came to light after 11 million confidential documents were leaked from the offices of law firm Mossack Fonseca, one of the leading experts and mediators on tax havens.
The documents were leaked to the International Consortium of Investigative Journalists, a Washington-based organisation.
Miguel Arias Cañete, European for energy and climate action commissioner, is the most prominent EU politician involved; his wife was named on the list.
The Russian revelations indicate that Russian leader Vladimir Putin's closest friend, Sergei Roldugin, a professional cellist whom he has known since his childhood, used Mossack Fonseca as well as Cyprus-based entities to create a web of financial interests worth up to €1.75bn.
Putin's spokesman Dmitry Peskov dismissed the claims as part of a coordinated plot against Russia, aimed at discrediting Putin prior to parliamentary elections earlier this week.
It is also alleged that Ukraine's president, Petro Poroshenko, set up a secret offshore company in the British Virgin Islands at a time when his troops were engaged in a bloody battle with Russian troops and pro-Moscow rebels.
The list of people implicated in Panama Papers also covers the late father of British Prime Minister David Cameron, the Spanish royal family, the government of former Greek Prime Minister Antonis Samaras, a former French minister, a Maltese energy minister, a former Warsaw mayor and a number of British MPs.
It also includes the names of 72 current and former heads of state and dictators accused of looting their own countries.
The documents also list footballer Lionel Messi, the former Iraqi president Iyad Allawi and the Prime Minister of Iceland, Sigmundur David Gunnlaugsson.
MEP reaction was swift, with S&D group leader Gianni Pittella saying, "This new tax scandal shows that we need to act more efficiently against fraudsters and against those who facilitate them in hiding their money in tax havens."
The Italian deputy added, "The fact that this is being used, not only by the criminal underworld, but by some political leaders confirms how widespread these immoral tax practices can go. It discredits the political system in the eyes of the citizens.
"We need to be more aggressive in the fight against tax fraud."
On Monday, he wrote to Commission President Jean-Claude Juncker asking him to "ensure that a large number of companies will be obliged to report publicly about their activities in every country in which they operate, be it inside or outside the EU, in order to strengthen the fight against aggressive tax planning by multinationals."
The MEP said, "The draft proposal currently under discussion on country-by-country reporting would fall short of the expectations and demands of Parliament and the citizens.
Further reaction came from Green tax spokesperson Sven Giegold, who said, "Panama Leaks shows we have so far just been scratching at the surface of the odious tax avoidance practises employed by individuals and businesses around the world and Europe. Almost a year and a half on from the ground-breaking 'LuxLeaks' revelations, we are still stuck in limbo as regards widespread tax avoidance.
"Authorities and policy makers in Europe and beyond are aware that individuals and corporations continue to go to great lengths to avoid paying their taxes but they are unwilling to take the necessary action to address this."
As well as the International Consortium of Investigative Journalists, the investigation was conducted by the German newspaper Süddeutsche Zeitung and more than 100 other news organisations around the globe.
The probe reveals the offshore links of some of the planet's most prominent people and, in terms of size, it is likely the biggest leak of inside information in history - more than 11.5 million documents stored in 2.6 terabytes of data.
In comparison, Wikileaks amounted at 92,000 documents while whistleblower Edward Snowden released around 15,000 documents.
The disclosures have led Transparency International EU to demand public registers of all companies' beneficial owners to make it harder for the corrupt to hide stolen assets in secret companies and trusts.
Carl Dolan, Director of Transparency International EU, said, "The Panama Papers leak reveals how the international finance system is used to allow the rich, powerful and corrupt to launder and hide stolen assets. Nothing short of public transparency about corporate ownership can stop the rot.
"There is a narrow window in the next six months for the EU to lead the way. It can then use its authority and experience to keep pressure on secrecy jurisdictions around the world to open up."