Hezbollah guerrillas captured two Israeli soldiers in the attack.
Israel responded with air and artillery strikes in south Lebanon, reports CBS News correspondent Robert Berger. Fighting has now spread to two fronts, after the kidnap of an Israeli soldier by Palestinians in Gaza two weeks ago.
Lebanese security officials said that six Israeli soldiers and two Lebanese civilians were killed. Israel said three of its soldiers were killed.
Israel's assault also continues in Gaza, where CBS News chief foreign correspondent Lara Logan reports an air strike targeting Hamas commanders missed, and instead killed a Palestinian family of nine — including seven children.
"As bodies were pulled from the rubble here, you could feel the crowd, the anger of the crowd," Logan said in an exclusive report from the scene. "As people surged forward, it almost seemed to go out of control and then, in one dramatic moment, a man in the crowd began a chant, a traditional Palestinian chant of resistance, also vowing revenge."
"Hundreds of people," Logan continues, "who in one moment, were surging forward in anger, almost out of control, suddenly stood rooted to the spot and repeated this chant, which has been a chant of resistance against the Israelis for the past few years."
Israel said it launched the air strike in the Sheik Radwan neighborhood, a Hamas stronghold, because Hamas commanders there were planning more attacks on Israel. Israel also said the home it attacked was a "meeting place for terrorists."
The Israeli army said its most wanted terrorist, Mohammed Deif was wounded, but Hamas denied it. Deif has narrowly escaped several assassination attempts in the past.
The military, which sent airplanes and gunboats to strike southern Lebanon, plans to call up a reserves division of several thousand soldiers, security officials said. Ground troops also entered to look for the soldiers and to keep them from being taken deeper into Lebanon, government officials said.
"Israel's response will be restrained but very, very, very painful," Olmert said.
Olmert said the attack was not an act of terror but an attack by a sovereign state on Israel. The Lebanese government, of which Hezbollah is a part, "must bear full responsibility," he said.
Hamas appeared to toughen its demands for the safe return of the soldier its militants seized two weeks ago, saying Hezbollah's actions strengthened Hamas' position.
"We have proven to this enemy (Israel) that the one option is the release of Palestinian, Lebanese and Arab captives. All captives, without exception," said Osama Hamdan, Hamas' spokesman in Lebanon.
"What happened has strengthened the issue of the captives, and the enemy will submit to our choice, which is the exchange of the captives in return for the release of the soldiers," he told Al-Jazeera television.
Hamas had previously demanded the release of some Palestinian prisoners in return for Cpl. Gilad Shalit's release.
Hamdan did not say whether Hamas had consulted with Hezbollah, but he said there may be subsequent "coordination and an understanding."
Hezbollah leadership has told CBS News that the capture of the two Israeli soldiers is yet further pressure to force Israel to release Palestinian detainees, reports Logan.
"We shall not surrender to arrogance and we will not negotiate with terrorists," Olmert said.
The nighttime aerial attack in Gaza was bound to intensify international criticism of Israel. The United Nations has already complained about what it said was disproportionate use of Israeli force in the Gaza operation.
But initially, the European Union demanded the immediate release of two Israeli soldiers captured by Hezbollah on Wednesday in a cross-border raid that provoked an Israeli strike in southern Lebanon.
EU spokeswoman Emma Udwin urged both sides to respect the Blue Line border drawn up by the U.N. after the withdrawal of Israeli forces from southern Lebanon in 2000.
"We're extremely concerned," Udwin said. "There has been a very serious incident."
She said the EU "unreservedly" condemned the kidnapping.
More than 60 Palestinians have been killed in Israel's Gaza offensive, most of them gunmen, but about a dozen have been civilians. One Israeli soldier also has died, shot by fellow troops.
Security officials said Israel dropped a 550-pound bomb on the home, which collapsed and buried people under the rubble.
The house next door, set on stilts, leaned precariously. All the windows in the surrounding houses were blown out, and some walls were pocked with small shrapnel holes. People sat in damaged houses, peering through holes in the walls.
The scene resembled the aftermath of a 2002 attack in which an Israeli warplane dropped on one-ton bomb on the house of a Hamas leader in Gaza, killing him and 14 other people, including nine children. The attack set off criticism from human rights groups that still reverberates.
Earlier in the day, Israel sent tanks and troops into southern Gaza from the Kissufim crossing, once the main access point to Jewish settlements, and an access road near city of Khan Younis and town of Deir al-Balah.