The UN peacekeeping force says Israeli soldiers were operating on their own side of the border when fighting broke out with Lebanese troops on Tuesday.
Clashes erupted when Israeli troops tried to cut down a tree that both sides claimed was in their territory.
Four people - including two Lebanese soldiers, a Lebanese journalist and an Israeli officer - were killed.
Israeli troops have returned to the area to finish the tree-clearing under Unifil's supervision, reports say.
At an emergency session on Tuesday, the UN Security Council expressed "deep concern" about the incident - the worst border clash for years between soldiers from Israel and Lebanon - and urged both sides to show the utmost restraint.
The leader of the Lebanese militant group, Hezbollah, said it would join the fighting if there were any more border clashes.
"The trees being cut by the Israeli army are located south of the Blue Line on the Israeli side," said Unifil spokesman Lt Naresh Bhatt in a statement, referring to a boundary drawn up by the UN in 2000.
It marks the agreed line to which Israeli forces withdrew after 22 years occupying southern Lebanon.
Lebanese Information Minister, Tareq Mitri told journalists that his country disputes the Blue Line demarcation in certain areas, including the village of Adaysseh, where the clash took place.
He insisted the location of the trees was "south of the Blue Line but in Lebanese territory."
Unifil accepted the Lebanese government had "some reservations" concerning the Blue Line in the area, but noted that both sides had committed to complying with the UN's decisions on the border.
Israel has welcomed the Unifil finding, saying it "clearly corroborates" the Israeli version of events.
"Our routine activity yesterday was conducted entirely south of the frontier - on the Israeli side - and the Lebanese Army opened fire without any provocation or justification whatsoever," said Mark Regev, a spokesman for Israeli Prime Minister Benjamin Netanyahu.
Israel said its troops were fired upon while conducting maintenance work along the border, and that UN forces had been notified in advance. Lebanon said its troops opened fire after an Israeli soldier breached the border fence.
Tensions are often high along the heavily fortified frontier, where Israel fought a 34-day conflict with Hezbollah in 2006.
Although Hezbollah did not take part in the latest violence, the group's leader, Sheikh Hassan Nasrallah, said his forces would not stand by idly if Israeli troops attacked Lebanese forces again.