The short video said the attack was carried out by the Mujahedeen Shura Council of Jerusalem, a murky group that was formed in April.
It identifies two men, one Egyptian and one Saudi, as the perpetrators of Monday's attack. "Soon we will carry out a double suicide mission against the enemy troops on the Egyptian border with occupy Palestine today, Monday, June 18," said the Saudi man.
Later Tuesday, the group issued a statement on an al-Qaida-linked website saying the men targeted an Israeli patrol with a bomb, anti-tank rockets and gunfire. It said the attack was dedicated to "Sheik Mujhahid Osama bin Laden," the al-Qaida founder who was assassinated by U.S. troops last year.
The Israeli military declined comment on the latest claims. Military officials have been warning for more than a year that al-Qaida is operating in the area.
Israeli officials believe the group has taken advantage of the power vacuum in the Sinai since the ouster of longtime Egyptian President Hosni Mubarak last year.
On Tuesday Israel's military chief, Lt. Gen. Benny Gantz, toured the area of the infiltration and said his forces are dealing with the situation.
"The more significant issue is what is happening inside Sinai, the dispatch areas, the terror bases that are expanding and growing there. Egypt must exercise its sovereignty in Sinai," he said.
Israeli officials also believe al-Qaida and other "global jihad" elements have infiltrated the Hamas-ruled Gaza Strip, which lies between Israel and the Sinai.
Hamas denies al-Qaida is operating in Gaza. The two militant groups have far different ideologies. Hamas says its struggle is focused solely against Israel, while al-Qaida claims to be fighting a holy war against the West.
Even so, Israel blames Hamas for all violence emanating from Gaza. Following Monday's attack, Israel carried out a series of airstrikes on militant targets in Gaza, drawing retaliatory rocket fire.
Two Palestinian men in Gaza were killed early Tuesday following an airstrike, which Israel said was in response to rocket fire. In all, at least 20 rockets fell on southern Israel Tuesday. Residents in the southern town of Sderot were told to stay close to shelter throughout the day, while a local college was forced to cancel its graduation ceremony. There were no reports of injuries on the Israeli side.
The flare-up in violence is the most serious fighting between Israel and Gaza militants in months, and has threatened an informal cease-fire that has largely held for the past three years. Even Hamas, which has largely stayed out of violence, claimed responsibility for some of the rocket fire.
In another development Tuesday, vandals torched and scrawled graffiti on a Palestinian mosque in the West Bank, Israeli security officials said. Suspicion fell on radical Jewish settlers angry over the looming demolition of an unsanctioned settler enclave.
By July 1, the government has committed to destroying 30 apartments built illegally on privately owned Palestinian land. Acts of vandalism against Palestinian property have been expected ahead of that date because radical settlers routinely attack Palestinian targets in retaliation for government settlement policy they oppose.
The Hebrew-language graffiti spray-painted on the mosque in Kfar Jabaa read, "Ulpana War" and "Price Tag," the names of the outpost and the settler retaliation campaign.
Prime Minister Benjamin Netanyahu condemned the attack and said authorities would act swiftly to bring the vandals to justice. "This is an act of intolerant and irresponsible lawbreakers," he said.
Police said no suspects had been arrested in the predawn attack.
Israeli construction on lands the Palestinians claim for a future state is at the heart of the current peacemaking deadlock.