The White House was keen to "make progress on all tracks in the peace process and to pursue regional peace and stability," he told reporters after meeting Assad.
"There are challenges on the road but my meeting with President Assad leaves me hopeful that we can make progress together in the interest of both our countries.
"I had quite productive and extensive discussions with President Assad. We talked about areas on which we disagree but also we found areas of common ground on which we can build," Burns said.
Burns said a member of his delegation, Dan Benjamin, the US State Department's coordinator for counterterrorism, was to stay behind for a day to hold more talks in Damascus.
According to Seymour Hersh of New Yorker magazine, cooperation has been restored between the CIA and Syria's intelligence services.
Assad, for his part, reiterated to Burns "the importance of the US role in the peace process" and renewed calls for Washington "to adopt a policy which pushes Israel to accept the demands for peace," state news agency SANA said.
Washington's ties with Damascus had been strained by Syria's three-decade-old alliance with Iran and US allegations of meddling in the affairs of its eastern neighbour Iraq.
Syrian support for Lebanon's Shiite militant group Hezbollah and the Palestinian Islamist movement Hamas which rules Gaza have also proved a stumbling block.
"I have no illusions about the challenges on the road ahead," said Burns.
Career diplomat Robert Ford will be the first US ambassador to Damascus since Washington recalled its envoy after Lebanon's former prime minister Rafiq Hariri was killed in February 2005 in a bombing blamed on Syria.
"If confirmed by the Senate, ambassador Ford will engage the Syrian government on how we can enhance relations, while addressing areas of ongoing concern," said White House spokesman Robert Gibbs.
But the nomination ignited a festering row with opposition Republicans over Obama's signature policy of seeking to engage governments considered hostile to Washington.
Ileana Ros-Lehtinen, the top Republican on the House of Representatives foreign affairs committee, criticised the move as "reckless engagement" and a reward for an enemy of the United States.
Obama has seen his efforts to engage Iran and broker peace between Israel and the Palestinians founder in his first year in office, and the overture to Syria may be aimed at seeking a new way to ease the deadlock.
But analysts say it seems unlikely that Assad's government, with a first priority of ensuring its own survival, will be keen to sever links with Iran or make immediate concessions to Israel.
US officials may be keen to increase intelligence cooperation with Syria, although its stakeholding in Lebanon through Hezbollah is likely to prove a long-term impediment to better ties.
Obama apparently paved the way for the announcement about Ford on Friday, calling Lebanese Prime Minister Saad Hariri to tell him he strongly supports the effort to bring the killers of his father to justice.
The previous administration of president George W. Bush recalled the US ambassador from Damascus and put relations with Syria on hold in 2005, following Rafiq Hariri's murder.
His death in a massive bomb blast on the Beirut seafront in February of that year was widely blamed on Syria, although Damascus has strongly denied any involvement.