The issue left London up against Paris, which would have to pay for the bulk of the shortfall otherwise, the diplomat said. He spoke on condition of anonymity because he wasn't allowed to discuss the closed-door talks publicly.
In the end, Van Rompuy said the British concerns were taken on board, since "actually nothing has changed" on the question of Britain's contribution since the February agreement.
French President Francois Hollande said he signed off on the deal and praised the European Parliament for winning more wiggle room on the budget.
The summit was initially meant to focus on finding ways to get more young people employed, and calmly taking stock of EU efforts to stabilize the world's biggest economic bloc now that its deep debt troubles have subsided.
Crucially, the EU budget also includes money for the employment measures that the bloc's leaders addressed at the two-day summit which finishes Friday afternoon. No budget agreement would mean no money for those projects.
Unemployment is at a record high of 11 percent for the EU and 12.2 percent for the 17 member countries that use the euro. It is far worse for the young: Latest figures show almost one in four people aged under 25 in the EU are unemployed. In Greece and Spain, that rate has it hit more than 50 percent.
After the late-night meetings, Hollande said that 6 billion euros for youth jobs will be speeded up and spent over 2014-2015 instead of over 7 years.
In addition he said that there will be two to three times that amount in "European credits" for employment schemes.
Germany argues that governments should focus on reforms instead of new funding, to get growth going again and create more jobs.
Thursday's deal on the budget came hours after EU finance ministers reached a landmark deal determining that banks' shareholders, creditors and holders of large deposits will have to bear the brunt of future bank failures, so that taxpayers don't have to.