The attackers spent months digging the tunnel, right under the nose of the army, reports CBS News Correspondent Robert Berger.
The attack was well timed - coming amid hopes on both sides for reviving the peace process following the death of Yasser Arafat.
Israel plans to pull out of Gaza next year, and it wants to coordinate the move with a moderate Palestinian leadership. The question is whether the Palestinian moderates can control the militants who want to torpedo the peace process.
Several structures at the crossing collapsed and others were damaged by the force of the blast.
Then gunmen opened fire on the base, and there was a second explosion an hour later. Hamas and another militant group, the Fatah Hawks, claimed responsibility.
Israel said the attack jeopardizes peace moves and demanded Palestinian action to stop the militants.
The military said one soldier was missing and was apparently trapped under rubble.
The attack was another sign that a lull in violence that followed Arafat's death on Nov. 11 is over. On Tuesday, an Israeli soldier was killed in a blast at the entrance to another tunnel near the Gaza-Israel border, setting off Israeli retaliation that killed four Palestinians.
Palestinian mortar and rocket barrages have hit Jewish settlements in Gaza daily, and militants have resumed firing homemade Qassam rockets at Israeli towns just outside Gaza. Israeli forces returned fire, wounding several Palestinians.
In another development, imprisoned Palestinian uprising leader Marwan Barghouti declared in a letter Sunday that he would throw his support to mainstream candidate Mahmoud Abbas in a Jan. 9 election to replace Arafat, dropping out of the race.
Israeli army spokesman Capt. Jacob Dallal said two explosions rocked the border terminal while it was open. "This was a very large, well coordinated, planned attack against an international crossing, used by Palestinian civilians to cross into Egypt."
Dallal said the crossing would be closed until further notice.
Israel TV defense analyst Yoav Limor called the blast the result of an "intelligence failure."
Palestinians said one gunman was killed in the exchange of fire after the blast, and a civilian was also killed, they said.
Masked Hamas militants said a gunman tried to kidnap a wounded soldier but killed him because the soldier resisted.
A Palestinian militant giving his name only as Abu Majad said the explosives-filled tunnel was 900 yards long. He said the attack was retaliation for what he called "the assassination" of Arafat, who died in a French hospital. Some Palestinians claim he was poisoned by Israel.
Gissin said the attack could jeopardize efforts to normalize Israeli-Palestinian relations and restart peace moves after Arafat's death.
"Unless there is decisive and sustained effort taken to dismantle the terrorist organizations, it will be impossible to move toward normalization and toward political negotiations," he said, demanding action by the Palestinian Authority.
Hours before the attack, Israel's Cabinet decided in principle to release 100 to 200 Palestinian prisoners as a gesture to Egyptian President Hosni Mubarak for Egypt's release last week of Azzam Azzam, an Israeli Arab who served eight years in prison on espionage charges.
Gissin said none of the prisoners to be freed were linked to fatal Palestinian attacks.
Palestinian officials said Israel should be coordinating prisoner releases with them. "Usually, the unilateral releases have never been satisfactory," Cabinet Minister Saeb Erekat said.
An official in Sharon's office, speaking on condition of anonymity, said the Gaza attack would not affect the prisoner release.
In Ramallah, Barghouti's associates called a news conference to read a letter from his prison cell. The letter was harshly critical of the Fatah leadership, but Barghouti threw his support to Abbas.
Ahmed Ghneim, who read out the letter, said Barghouti would drop out of the race.
Barghouti's wife, Fadwa, was to formally withdraw his candidacy on Monday.
In the letter, Barghouti made a list of demands of Abbas. He rejected moves to disarm militant groups — a key Israeli demand — and said no agreement should be made without release of all prisoners. He did not present the demands as conditions for his support.
Israel has said Barghouti, serving five life terms after convictions in deadly Palestinian attacks, will not be freed.
Barghouti was the West Bank leader of Fatah when he was captured by Israeli forces in 2002. Since Fatah nominated Abbas, Arafat's longtime deputy, as its candidate, Barghouti registered to run as an independent, drawing harsh criticism from Fatah faithful for splitting the party.
Polls last week showed Barghouti and Abbas running a close race.
In Israeli politics, meanwhile, teams from Sharon's Likud Party were negotiating with the moderate Labor and ultra-Orthodox Jewish Shas parties to expand Sharon's shaky coalition to enable implementation of Sharon's Gaza pullout plan next year.