A senior Israeli officer was killed in a confrontation with militants on the frontier with Lebanon on Monday, Israel's military has confirmed.
The military said troops and aircraft killed two "terrorist infiltrators" who had crossed into Israeli territory.
Palestinian Islamic Jihad claimed that they were its members.
Later, at least three militants from Lebanon's Hezbollah movement were killed as the military attacked sites in Lebanon in response to mortar fire.
On Tuesday, Hezbollah said its fighters fired an anti-tank guided missile at an Israeli military vehicle in the Avivim area.
The military said it caused no casualties and that a helicopter attacked one of the group's observation posts in response.
The Palestinian militant group Hamas also claimed that its fighters in Lebanon had fired rockets towards Israel's western Galilee region.
The rockets were intercepted or fell in open areas, according to the Israeli military, which retaliated by shelling another two Hezbollah observation posts.
It was the third day in a row that attacks have been launched from Lebanese territory, and it coincided with a war between Israel and Hamas in Gaza.
The US has warned Hezbollah, which is Lebanon's most powerful military force and fought a war with Israel in 2006, not to get involved.
Monday's deadly violence began when a number of armed men crossed the frontier near the Lebanese town of Dhayra and were killed by Israeli troops, supported by helicopter gunships, the Israeli military said.
Two of the militants were killed and one escaped back to Lebanon, it added.
Palestinian Islamic Jihad said the infiltration was part of "Operation Al-Aqsa Flood" - the name given by the militant group Hamas for the unprecedented assault from Gaza on southern Israeli communities that began on Saturday, in which at least 900 Israelis are reported to have been killed.
The Israeli military also said its helicopters attacked Hezbollah positions inside Lebanon, including two observation posts, after two mortar bombs were fired into Israel but caused no casualties.
Hezbollah later announced that three of its fighters were killed in Israeli attacks on Lebanese towns and villages. The group said it responded by firing rockets and mortar shells at two Israeli military barracks.
On Sunday, Hezbollah fired a number of shells and rockets at Israeli military positions in the disputed Shebaa Farms/Mount Dov area in what it called an act of "solidarity with the Palestinian resistance".
The Iran-backed group, which has long range rockets that could strike deep into Israeli territory, has threatened to intervene in the Israel-Gaza conflict if certain "red lines" are crossed. They reportedly include an Israeli invasion.
A senior US defence official told reporters on Monday: "We are deeply concerned about Hezbollah making the wrong decision and choosing to open a second front to this conflict."
