British newspapers said the choices would not help the EU to achieve a greater global impact and questioned whether Baroness Ashton would carry much weight in her new role, which had been dubbed the "EU's foreign minister".
But Lady Ashton, the former EU trade commissioner, said European leaders were "very comfortable" with her appointment.
"Over the next few months and years I aim to show I am the best person for this job," she said. "I think for quite a few people, they would say I am the best for the job and I was chosen because I am."
The post of president has become a much less important one than originally envisaged, our correspondent adds.
German Chancellor Angela Merkel said Mr Van Rompuy would bring consensus and political competence to the presidency.
However, Turkish member of parliament Onur Oymen said he was concerned about what Mr Van Rompuy's presidency could mean for Turkey's aspirations to join the EU.
He told the BBC's World Today that Mr Van Rompuy had "said a few years ago he was totally against Turkish membership because of religious and cultural reasons".
"We are not very optimistic about the future of our relations during his presidency," Mr Oymen said.
Mr Van Rompuy said he would be "discreet" in his new job, adding that he sees climate change and Europe's high unemployment as key concerns in the years ahead.
Baroness Ashton and Mr Van Rompuy were chosen unanimously by the 27 EU leaders at a summit in Brussels.
Both have been seen as consensual politicians with limited foreign policy experience.
US President Barack Obama said the appointments would "strengthen the EU and enable it to be an even stronger partner to the United States".
He said the US had "no stronger partner than Europe in advancing security and prosperity around the world".
'Milestone for Europe'
US Secretary of State Hillary Clinton said the appointments were a "milestone for Europe and for its role in the world".
Herman van Rompuy: Every country should emerge victorious from negotiations
She said she was looking forward to working closely with Mr Van Rompuy and Baroness Ashton on issues of mutual concern, including the Iranian nuclear debate, achieving stability in Afghanistan and promoting a peace agreement in the Middle East.
The President of the EU Commission, Jose Manuel Barroso, also praised the appointments, saying it would be "impossible to find a better choice than those personalities for the European Union leadership".
Mrs Merkel said of Mr Van Rompuy: "We got a candidate who brings consensus and whose political competence have long been tested and tried throughout his political career."
French President Nicolas Sarkozy said it had been "a very wise decision" to choose a candidate from "an important country but not one of the most important countries, so that no-one will feel excluded" from EU debate.
Addressing the summit after his appointment, Mr Van Rompuy said the EU was "living through exceptionally difficult times".
THE PRESIDENT'S ROLE
Chosen by leaders of 27 member states
Can be re-elected once
Chairs EU summits
Drives forward the work of EU Council of Ministers
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.