Ahead of the trip, Mr Feltman said the US envoys had a "long list" of concerns to discuss with Syrian officials.
Mr Feltman said: "Our trip to Syria... is an opportunity for us to start addressing these concerns and using engagement as a tool to promote our objectives in the region."
He also sought to reassure the Lebanese that US support for Beirut was "unwavering", despite the latest overtures towards Damascus.
The pair are expected to return to Beirut on Sunday for further consultations.
BBC state department correspondent Kim Ghattas, in Washington, says the US is concerned by Syria's support for militant groups such as Hamas and Hezbollah, alleged covert nuclear activities and interference in Lebanon's affairs.
But Mr Feltman said the talks were not about finger-pointing but about discussing differences and identifying areas where American and Syrian concerns overlapped.
He said Washington's strategy for engagement was goal-oriented, a way to achieve objectives for diplomacy, our correspondent adds. No benchmarks or dates for further meetings were set.
Last month several leading US congressmen, including Senator John Kerry, went on an unofficial visit to Damascus for talks with the president.
'Gateway to peace'
Mr Feltman served as US ambassador in Beirut until 2008, and is currently an acting assistant secretary of state.
As a congressional aide, Mr Shapiro was instrumental in drafting the Syria Accountability Act, which placed sanctions on Damascus in 2003.
The killing of Rafik Hariri in 2005 sparked outrage in Lebanon
During his time in Lebanon, Damascus accused Mr Feltman of orchestrating an anti-Syrian movement in Lebanon.
Syria's ambassador to Britain, Sami Khiyami, told the BBC before Saturday's meeting: "Syria is the gateway to peace in the Middle East.
"I think the American government, Mr Obama personally, has to take care of the aspirations of the Arab peoples and not only of the Israeli people."
The last senior US official to visit Damascus was Richard Armitage, then deputy secretary of state, in January 2005.
Washington withdrew its ambassador from Syria shortly afterwards, following Mr Hariri's assassination.
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