Official statistics show EU migration added 74,000 to the UK population in the year to June.
The figure is the lowest since the year ending September 2012, when it stood at 65,000.
However, the UK Government remains far from meeting is overall migration target of 100,000 with non-EU net migration at its highest since 2004.
248,000 more non-EU citizens arrived than departed in the year to June, the Office for National Statistics (ONS) said. Overall, net long-term international migration was 273,000.
Nicola Sturgeon said the figures were “not good news” for Scotland, and Labour attacked the Government for driving away workers needed to fill shortages in professions like nursing and care.
The figures come as Parliament awaits the Government’s much-delayed proposals for Britain's post-Brexit immigration system.
However, addressing the Commons Liaison Committee on Thursday morning, Theresa May did not give an assurance that the plan would be published before MPs vote on the Brexit deal on 11 December.
"We are still in discussion as to the date as to when the immigration white paper will be published," she told the committee.
Jay Lindop, director of the ONS's centre for international migration, said: "Net migration continues to add to the population and has remained fairly stable since its peak in 2016.
"However, there are different patterns for EU and non-EU migration.
"Due to increasing numbers arriving for work and study, non-EU net migration is now at the highest level since 2004.
"In contrast, EU net migration, while still adding to the population as a whole, is at the lowest since 2012."
For the second consecutive quarter, the figures showed more citizens of eight eastern European countries left the UK than arrived.
Net migration from the so-called EU8 nations which joined the bloc in 2004, including Poland and the Czech Republic, was minus 14,000 in the year to June.
The ONS report said: "This has been driven by a decrease in EU8 immigration, particularly for work, and an increase in emigration over the last two years.
"There could be several reasons for this changing pattern, decisions to migrate are complex and people's decision to move to or from the UK will be influenced by a range of factors."
Nationals of 14 longer-term EU member states, such as France, Germany, Italy and Spain, continue to add to the UK population with net migration of 47,000.
An estimated 34,000 more Romanians and Bulgarians arrived than left, although this figure has almost halved since the Brexit vote in 2016.
While immigration data published since the EU referendum have sparked claims of a "Brexodus", figures for the rest of the world are tracking in the opposite direction.
Non-EU net migration showed a "statistically significant" year-on-year rise, reaching almost a quarter of a million.
While the UK remains bound by EU free movement rules, immigration from the rest of the world is subject to restrictions.
Shadow home secretary Diane Abbott said: "These data reinforce the view that the Government's net migration target is reckless and foolish. The target has never once been met and non-EU migration alone far outstrips it.
“The truth is international students provide a huge benefit to this country and we have shortages of doctors, nurses, care workers and many more. Yet this government campaigns to slash these numbers.
“If that is the content of their new immigration Bill, whenever they have finished fighting over it, we will all be worse off.”