Newly-appointed Secretary of State Hillary Clinton described the Middle East and Afghanistan-Pakistan as the two most pressing foreign policy issues confronting the US.
"We have no time to lose," said Mr Obama after Mrs Clinton introduced the new envoys.
Mr Mitchell would head to the Middle East as soon as possible to shore up the fragile truce between Israel and Hamas in Gaza.
Israel must re-open the border crossing into the Gaza Strip "to allow the flow of aid and commerce," Mr Obama said.
"Hamas must end its rocket fire... The United States and our partners will support a credible anti-smuggling and interdiction regime so that Hamas cannot re-arm," he added.
But Hamas spokesman Osama Hamdan told al-Jazeera television that Mr Obama would fail in the Middle East unless he changed his position.
Israel ended a three-week offensive on Sunday, pounding the Gaza Strip with bombs and shells and sending in tanks and infantry in an attempt to stop Palestinian militants firing rockets into southern Israel.
Richard Holbrooke, a former ambassador to the UN and the architect of the Dayton Accord that ended the war in the Balkans, is to co-ordinate US policy in Afghanistan and Pakistan.
Mr Obama said there was a "deteriorating situation in Afghanistan and Pakistan" and that the region was now "the central front" in the battle against extremism.
The office of Afghan President Hamid Karzai said he spoke to Hillary Clinton on Thursday to discuss the "joint struggle against terrorism and relations between the two countries".
The Afghan president emphasised to Mrs Clinton "that Afghanistan is a friend of the United States", a statement said.
Earlier, as he signed orders to set in motion the closure of Guantanamo Bay, Mr Obama said the US would continue to fight terror, but would maintain American values while doing so.
"The United States intends to prosecute the ongoing struggle against violence and terrorism," he said.
"We are going to do so vigilantly, we are going to do so effectively, and we are going to do so in a manner that is consistent with our values and our ideals."
Mr Obama has repeatedly promised to close the Guantanamo Bay prison, where some 250 inmates accused of having links to terrorism remain. Many of the detainees have been there for years and have never been charged.
Mr Obama said American values would be maintained in foreign policy
The move was welcomed by President Karzai, who has repeatedly called for detained Afghan citizens to be released so they can face trial in Afghanistan.
"This decision by the United States is a major step towards bringing more international support to the struggle against terrorism, and enlisting all nations in this war," he said.
However, correspondents say closing the prison will not be easy. Questions remain over where those charged will be tried and where those freed can be safely sent.
Secret CIA prisons around the world are also to be closed, although the time frame for this is unclear.
The secret rendition - or transfer - of terror suspects to these prisons was widely criticised after they came to light in the wake of the invasion of Iraq in 2003.
Mr Obama also banned the use of threats, coercion, physical abuse and waterboarding in the interrogation of suspects - all techniques that had been permitted under the Bush administration.
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.