"For us, that's victory. That's what we wanted," he told CNN's Wolf Blitzer.
Israel will hold Hamas responsible for any attacks from Gaza, whether conducted by that organization or any others, Regev said. He said the agreement reflects that understanding.
Regev also said the deal calls for immediate talks on economic restrictions on Gaza.
"If the border is quiet, that enables us to be more forthcoming," he said. "The arrangements agreed with the Egyptians say we'll start talking from tomorrow about a process to work on those issues."
(12:49 p.m. ET, 7:49 p.m. local) U.S. President Barack Obama on cease-fire announcement
President Barack Obama on Wednesday commended Israeli Prime Minister Benjamin Netanyahu for following his recommendation to agree to a cease-fire with Hamas that was brokered by Egypt, the White House said.
(12:41 p.m. ET, 7:41 p.m. local) Hillary Clinton on cease-fire announcement
The Israeli-Hamas cease-fire announced Wednesday should "improve conditions for the people of Gaza and provide security for the people of Israel," U.S. Secretary of State Hillary Clinton said.
She made the remark at the news conference at which Egypt's foreign minister announced the cease-fire would begin at 9 p.m. local time (2 p.m. ET) Wednesday.
(12:30 p.m. ET, 7:30 p.m. local) Cease-fire to take effect Wednesday night, Egypt's foreign minister says
A cease-fire between Israel and Hamas will take effect at 9 p.m. local time (2 p.m. ET) Wednesday, Egypt's foreign minister announced in Cairo.
(12:15 p.m. ET, 7:15 p.m. local) Hamas leader Khaled Meshaal on the Tel Aviv bus attack
CNN chief international correspondent Christiane Amanpour, in an interview in Cairo, asked Hamas leader Khaled Meshaal whether Hamas was responsible for Wednesday's bombing of a bus in Tel Aviv, Israel.
"Not Hamas, not others, not other people from, not Hamas. No one can announce except those who committed (it), not me," Meshaal said. "The lesson is what matters. What led to this? Who created the circumstances that led to this (operation)? It is (Israeli Prime Minister) Benjamin Netanyahu with his crimes, in killing the kids of Gaza, and the continuity of aggression. He (creates) such ramifications everywhere. This could lead to any kind of reaction as retaliation for what happened in Gaza."
(9:25 a.m. ET, 4:25 p.m. local) Israeli President Shimon Peres on chances for a cease-fire
CNN's Wolf Blitzer asked Peres if there would be a cease-fire. "I hope so," Peres replied. "It is not yet done. There are difficulties, in any negotiation there are difficulties, but this time their expectations are over the horizon. But we are waiting and trying. The situation is very strange; both sides won't like a ground operation, neither us or them. A cease-fire is preferred by both sides, but the conditions they start to put (out) are little bit exaggerated."
When asked what Hamas wanted of Israel, he added: "They want Israel to do nothing. We shall not fly over, we shall not guard the border. Many others. The conditions of Israel (are) basically security conditions. Their conditions are political ones, and this is a contradiction."
(8:43 a.m. ET, 3:43 p.m. local) State Department on Hillary Clinton talks with Mahmoud Abbas in the West Bank
State Department spokeswoman Victoria Nuland issued the following statement: