Presidents Assad and Sleiman "instructed their foreign ministries to take the necessary measures in this regard to conform with the laws of the two countries," a statement read.
Syria and Lebanon have not had diplomatic ties since independence from French colonial power about 60 years ago, but Assad and Sleiman agreed to establish relations during talks last month in Paris.
Wednesday's confirmation of the decision marks the first fruits of a landmark trip to Damascus by Sleiman, the first Lebanese president to visit Damascus since Syria withdrew its troops from Lebanon in April 2005, ending almost three decades of military domination of its "sister" nation.
The Syrian pullout came two months after the assassination in a bomb blast in Beirut of Lebanese ex-prime minister Rafiq Hariri, for which Damascus has denied any responsibility despite accusations by Lebanese anti-Syrian groups.
Sleiman, a former army chief elected by parliament in May, was given a red-carpet welcome by Assad at the People's Palace overlooking Damascus on his arrival.
"[Assad] has instructed all concerned Syrian officials that he wants this visit to be successful and fruitful," Walid Muallem, the Syrian foreign minister, said in comments published in Lebanon's As-Safir newspaper.
"The visit is a starting point ... for future relations. We hope it will yield good relations in the interests of the Lebanese and Syrian people".
The agenda features issues such as a border demarcation, a review of longstanding accords, Lebanese detainees in Syria and the presence of pro-Syrian Palestinian groups in Lebanon, as well as the opening of embassies.
Sleiman's visit aims to redefine ties between Beirut and Damascus which have been on the decline since the al-Hariri murder.
It came a day after Beirut's Western-backed government won a vote of confidence in parliament, after stormy debates on the issue of weapons held by the Syria-backed Hezbollah group.
The vote allows the 30-member cabinet formed a month ago by prime minister Fouad Siniora to finally start work.