The Office for National Statistics said this morning that 24,000 citizens from the two countries arrived in the year to September 2013, nearly three times the 9,000 who arrived in the previous 12 months.
The ONS said this was “statistically significant” and that around 70 per cent came to work, while 30 per cent came to study.
Their figures also showed the Government was slipping behind its target for reducing overall net migration–the difference between migrants leaving and arriving in the UK.
Figures are not yet available for the numbers of Romanians and Bulgarians who have arrived in the UK since January 1 when labour market restrictions were lifted to comply with EU rules.
Ukip leader Nigel Farage said: "The immigration targets were nothing more than spin to appease in the short term the public who are concerned about uncontrolled immigration.
"But how can you have any targets when you can't control who comes to live, settle and work in this country? It's as sensible as hefting butterflies."
Despite the sharp rise, Immigration and Security Minister James Brokenshire said a dip in the number of migrants arriving from outside the EU showed the Government was getting to grips with the issue.
He said: "Our reforms have cut non-EU migration to its lowest level since 1998 and there are now 82,000 fewer people arriving annually from outside the EU than when this government came to power.
"And overall figures are also well down from when we first came to government in 2010 — with nearly 70,000 fewer migrants coming to the UK. Numbers are down across the board in areas where we can control immigration, but arrivals from the EU have doubled in the last year."
Mr Brokenshire admitted the limitations of the Government's ability to restrict immigration from the EU.
A Home Office spokesman said reducing net migration to the tens of thousands "remains the government’s objective".