The incidents follow Syria's capture on Saturday of five fishermen - four Lebanese and a Syrian - in Syrian territorial waters and an ongoing border dispute that has been delaying hundreds of trucks leaving Lebanon into neighbouring Syria bound for other Arab countries.
Many Lebanese speculate that the Syrian measures are in retaliation for April's forced withdrawal of thousands of Syrian troops from Lebanon amid the maelstrom that followed the 14 February assassination of former Lebanese prime minister Rafiq al-Hariri.
Syria was blamed in many Lebanese quarters for having a role in the slaying, but Damascus has repeatedly denied any such involvement.
Arab League Secretary-General Amr Moussa, who is expected to meet Syrian President Bashar al-Assad in Damascus on Sunday, has called for both sides to maintain good relations.
Sunday's shootout took place just outside the Lebanese village of Qaa, 100km northeast of Beirut, in an area where Lebanon's border with Syria is not clearly demarcated.
Lebanon recently increased its patrols along the area's mountain trails to curb smuggling.
Amr Moussa is to meet Bashar
al-Assad in Damascus on Sunday
The smugglers had infiltrated Lebanon from Syria and were returning with a load of unspecified contraband when they were spotted and fired at by a Lebanese customs patrol, a border policeman said.
One Lebanese officer was slightly wounded and the smugglers crossed back into Syria, the official said.
The Lebanese border policeman also said a Syrian border patrol member shot at Lebanese customs officers.
Syrian customs officials said they had not heard of the border shootout.
Security officials in northern Lebanon, speaking on condition of anonymity because of security issues, said a Syrian sea patrol arrested four Lebanese fishermen on Sunday after they entered Syrian territorial waters at Aridah, north of the Lebanese city of Tripoli.
On Saturday, Syria captured two Lebanese fishing boats and four Lebanese and a Syrian fisherman for fishing in the same area.
Syria has reportedly shut down
eastern Lebanon's Masnaa border
Many Lebanese opponents of Damascus celebrated after Syria - under US and UN pressure - was forced to withdraw thousands of military and intelligence forces from Lebanon, where they had been based since the early stages of Lebanon's 1975-90 civil war.
Moussa, who arrived in the Syrian capital on Saturday, urged Lebanon and Syria to continue to base their "very special" relationship on a "positive attitude by both sides".
Meanwhile, the Christian Voice of Lebanon radio station said on Sunday that Syrian authorities had completely closed eastern Lebanon's Masnaa border crossing on Saturday, even refusing entry for trucks with foreign licence plates.
Syrian troops were pressed into
pulling out from Lebanon in April
Lebanese truckers have been complaining for nearly three weeks of new, tough measures imposed by Syrian customs officials at Masnaa and on the Abboudieh-Dabbousieh crossing in the north.
Hundreds of long-haul trucks with perishable produce and other merchandise destined for Gulf countries have been backed up for kilometres on the Lebanese side pending long searches and border formalities by Syrian customs and security officers.
Syria contends the searches are required by security considerations. It has refused to speed up the process.
Damascus had long been the main powerbroker in Lebanon and the military withdrawal, coupled with the anti-Syrian outpouring in Beirut following al-Hariri's death, came as damaging blows to the bigger neighbour.