The clash occurred on Wednesday hours after Israel dropped thousands of anti-Hizb Allah leaflets over Lebanon, and two days after a cross-border shootout that saw some of the heaviest fighting in years.
Also on Wednesday, the UN Security Council expressed "deep concern" over the fighting and said Hizb Allah started it.
After prodding by the United States, the council statement deleted any reference to Israel by name.
"I wanted to underline the importance of saying expressly that it was Hizb Allah that initiated the attack," US Ambassador John Bolton told Reuters.
Hizb Allah's Al-Manar television said the clashes began when fighters engaged Israeli troops who had crossed into Lebanon.
Aljazeera reporter in Lebanon Abbas Nasir said the Israeli hang-glider pilot fell near a landmine field in Lebanese territory, a few metres from the border.
Hizb Allah fighters rushed to the area to arrest the Israeli, he said. Israeli forces opened fire on the Hizb Allah fighters, preventing them from reaching the pilot.
Israeli soldiers are posted along
the border with Lebanon
Israeli warplanes fired on the area, and an Israeli force rescued the pilot, Nasir said.
Israeli security officials said their troops did not enter Lebanon.
The Israeli troops fired on Hizb Allah and opened a gate in the border fence to allow the hang-glider pilot to run back into Israel, the security officials said, speaking on condition of anonymity.
Lebanese security officials, who also spoke on condition of anonymity, said the hang-glider must have been blown off course by the wind.
They said the incident occurred in the Meiss el-Jabal area.
A Security Council statement drafted by France on Monday condemned "military exchanges initiated by Hizb Allah" as well as Israeli violations of Lebanese air space.
The United States wanted the reference to Israel dropped, and Algeria, the only Arab member of the council, objected to pinning the blame on Hizb Allah, diplomats said.
US envoy John Bolton said
Hizb Allah started the clashes
In the end, members of the Security Council expressed "deep concern about the hostilities, which were initiated by Hizb Allah".
"They appealed to all parties to respect the (boundary) in its entirety, to exercise utmost restraint and to refrain from any action that could further escalate the situation."
The statement also called on the Lebanese government "to extend its authority and to exert its monopoly over the use of force all over its territory".