"It's not about gathering power, it's about saying as a team we are much more powerful and I will represent that team."
Downing Street said Lady Ashton's appointment was "a very good result for Britain" and dismissed suggestions she was not up to the job.
The prime minister's official spokesman also rejected suggestions her past support for CND - she worked for the anti-nuclear group in the late 1970s - meant she was unsuitable for a post that involved responsibility for EU security policy, saying it was "a very long time ago".
The spokesman said Mr Brown had spoken to Mr Blair on Thursday after it became apparent that he was not going to get the presidency.
"Tony Blair, I think, completely understands the situation. I think he has been very understanding about how these meetings have panned out," he said.
Mr Brown said he still believed Mr Blair would have been "excellent" in the president's role but the Conservatives expressed satisfaction that the lower-profile Belgian, Mr van Rompuy, had been appointed instead.
Mr Brown has welcomed the decision to appoint his friend and ally Baroness Ashton to the new EU role, saying it gave Britain "a powerful voice in Europe".
BBC Europe editor Gavin Hewitt said some Eurosceptics would be celebrating the appointment of Mr van Rompuy because it signalled an end to calls for a powerful European Council president who would impact on the world stage.
Shadow foreign secretary William Hague said: "I am very pleased that those of us across Europe who said that the president should be a chairman, not a chief, have won the argument."
Baroness Ashton is ideal for the role. She has never had a proper job and never been elected to public office
Former Labour leader and former EU commissioner Lord Kinnock said she was a "very substantial" figure who would do an outstanding job, while Lord Mandelson said the appointment of Mr van Rompuy and Baroness Ashton would help ensure Europe punched above its weight.
But Nigel Farage, leader of the UK Independence Party, which campaigns for Britain's withdrawal from the EU, branded the decision "disgraceful".
"We've got the appointment of two political pygmies," he said. "In terms of a global voice, the European Union will now be much derided by the rest of the world."
He added: "Baroness Ashton is ideal for the role. She has never had a proper job and never been elected to public office."
The Liberal Democrats welcomed the news. Foreign affairs spokesman Ed Davey said: "With low-profile appointees, no-one can take seriously any longer the Eurosceptic deception that these positions would challenge the supremacy of nation states acting together when they agree."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.