"President Morsi says that this cowardly attack will not go without a response… and that those who committed this crime will pay dearly," said a statement issued late on Sunday by the Egyptian president’s office.
After killing the border guards at a checkpoint near the Karm Abu Salem border crossing, the militants, who were wearing traditional Bedouin clothing, seized two armored vehicles and quickly drove off the scene.
Witnesses said the militants also fired rocket-propelled grenades at the Egyptian army checkpoint.
Meanwhile, an official of the Islamic Resistance Movement Hamas declared that, "The border between Egypt and Gaza is protected. We closed the tunnels to prevent anyone from escaping (into Gaza) and we have put our forces on alert."
"We reject any idea that Gaza was involved in this incident and we send our condolences over the killing of Egyptian soldiers," he added.
In Israel, a spokesman for the armed forces said that one of the vehicles exploded while trying to cross the Rafah border. The other vehicle was reportedly hit by an Israeli airstrike.
Egypt has been trying to secure its borders following last year's revolution, which resulted in the ouster of Hosni Mubarak.
In January 2011, the Egyptians launched the revolution against the pro-Israeli regime, which eventually brought an end to the 30-year dictatorship of Mubarak.