There had been a proposal to ban Facebook, Twitter, Instagram and other services from processing personal data belonging to under-16s unless their parents had given consent.
That would have effectively prevented the companies from offering their services to young people.
Tech firms lobbied against the idea.
"Member states will now be free to set their own limits between 13 and 16 years," said Jan Philipp Albrecht, the European Parliament's lead MEP on the new data protection rules.
The news was welcomed by Conservative MEP Timothy Kirkhope, who said: "Concerns have been listened to and the UK's age of consent will not be forced to change."
The European Commission and European Parliament had originally proposed that there should be a uniform age of consent set at 13 years, which would have brought the EU in line with the US.
At present most of Europe already follows this rule, although Spain's age of consent is set at 14.
It emerged late last week, however, that a last minute amendment to new data protection regulations that had been proposed by some states said the age limit should be raised to 16.
That prompted a coalition of internet safety organisations, including the Family Online Safety Institute, to warn the move could backfire.
"We feel that moving the requirement for parental consent from age 13 to age 16 would deprive young people of educational and social opportunities in a number of ways, yet would provide no more (and likely even less) protection," they wrote in an open letter.
They also raised concerns that such a major shift in policy could occur without wider public consultation.
The end result is that the EU's effort to create a "single digital market" will be limited in this area.
The new draft law, allowing countries to set their own social media age limit, is set to be confirmed by a vote in the European Parliament's civil liberties committee on Thursday and then by a full parliament vote next year.
Source: BBC News