The Israeli government gave final approval today to the American-crafted plan to evacuate the trapped Palestinian guerrillas from Beirut and warned Syria that a "very dangerous situation" had developed in eastern Lebanon because of cease-fire violations by Palestinian forces operating from behind Syrian lines there.

After 75 days of war and siege of the Lebanese capital, the Israeli Cabinet unanimously accepted the evacuation plan with little discussion and without a formal vote.

In Beirut, the Lebanese government formally asked the United States, France and Italy to send the troops that are to oversee the evacuation. A vanguard of 350 French Foreign Legion paratroopers is expected Saturday.

Israeli officials said they expect the Palestine Liberation Organization fighters to begin leaving Beirut Saturday or Sunday and to complete their withdrawal within two weeks.

The officials said approval of the plan was conditioned only on the return before the evacuation begins of an Israeli pilot and soldier reportedly being held by the Palestinians and the bodies of nine soldiers missing from this war or the 1978 Israeli invasion of Lebanon.

"The road is open to make it happen with only one condition--that before it happens we get our people back," said Dan Meridor, the Cabinet secretary.

The officials indicated that Israel has firm assurances from U.S. special envoy Philip C. Habib , the chief architect of the evacuation plan, that the captives and the bodies of the soldiers will be returned, possibly Friday.

During the Cabinet meeting, officials said here, Israel sought and received from Habib a handful of minor changes in the final plan. The requests were relayed to Habib in Beirut by telephone through the U.S. Embassy in Tel Aviv.

The reported capture last night of an Israeli soldier by Palestinian guerrillas near the Burj al Barajinah refugee camp, just south of Beirut, added a last-minute complication to the plan, crafted over weeks of tortuous negotiations by Habib and Lebanese officials who acted as intermediaries. The soldier joined Lt. Aharon Ahiaz, a pilot shot down over southern Lebanon in the early days of the war, as the only Israeli captives of the PLO to be acknowledged here.

[PLO officials denied reports that the Israeli soldier was ambushed, Washington Post correspondent Loren Jenkins reported from Beirut. They said the Israeli wandered into their lines, apparently lost, and added that he would be freed Friday along with the pilot. However, there was no official PLO announcement about capture of the soldier.]

Israeli officials said there are certain risks to Israel in accepting the plan, such as the possibility that some of the Palestinian guerrillas will attempt to remain behind in Beirut. But they said Habib had pledged to "minimize" these risks. They refused to comment on a report here that Israel has asked the multinational force that is to oversee the evacuation to photograph each of the departing guerrillas.

The plan calls for all the PLO guerrillas as well as the members of the Palestine Liberation Army--Palestinian fighters who are attached to Syrian Army units -- to evacuate Beirut and leave Lebanon "without even stopping for a drink of water," one official said. However, the 1,500 to 2,500 regular Syrian Army troops also trapped in the city will be allowed to redeploy behind Syrian lines in the Bekaa Valley of eastern Lebanon under the plan, the official said.

The warning to Syria was issued after the Cabinet meeting by a senior Israeli official and came amid new charges by Israeli military authorities that there have been repeated, daily cease-fire violations emanating from the Syrian-controlled territory in eastern Lebanon.

"We want total quiet on the eastern front," he said. "In the last week, once again there has been a return to terrorist activity done at least with the knowledge of the Syrians. This is very serious. That front has to be kept quiet until we reach a political solution. It is a very dangerous situation, and if it continues it is clear that Israel will have to react."

The official said Israel has made these concerns known to American officials, who have been asked to use their influence with Damascus to have Syria restrain the PLO forces operating behind its lines.

The warning to Syria was a graphic reminder that the withdrawal of the Palestinian guerrillas from Beirut will not in itself bring peace to Lebanon.

Syria has approximately 30,000 troops in the Bekaa Valley, augmented by PLO guerrillas, and there are additional Palestinian fighters in northern Lebanon around the port city of Tripoli. Israel has vowed to continue its occupation of southern Lebanon until all of these forces also agree to leave the war-ravaged country.

Senior officials here made it clear that they expect a continued heavy American involvement in the so-called "second stage" negotiations on the withdrawal of these forces. "It will be very difficult to reach an agreement without the good offices of the United States," one official said.

Yehuda Ben-Meir, the deputy Israeli foreign minister, thanked the Reagan administration and Habib in particular today for their role in negotiating the Beirut evacuation plan. If President Reagan decided to keep Habib in the Middle East for the second stage of negotiations, he said, "we would welcome his participation."

Despite their expressed satisfaction over the details of the plan, Israeli officials clearly were irritated by statements by PLO Chairman Yasser Arafat that it actually represented a political victory for the Palestinians. "I wish Yasser Arafat such victories all his life," Begin said.

Another senior official said the PLO agreed to withdraw from Beirut only after being confronted with the stark choice of "leave or die."

"There is no bravery here, and no altruism," he said.

The official said Israel's war in Lebanon had achieved a number of objectives, including the removal of the PLO as a threat to northern Israel and as the "center of international terrorism" based in Beirut, the destruction of the PLO's military organization "that was waiting to participate in an Arab war on Israel," and the elimination of the PLO as a factor in internal Lebanese politics.

The official warned that in the case of future PLO acts of terrorism anywhere in the world, "we will try to hit them where they are." Asked if this meant Israel might bomb Damascus in retaliation for a terrorist attack -- as the Israelis bombed Beirut just before the outbreak of the war in response to the attempted assassination of Israel's ambassador to Britain -- the official replied:

"It doesn't mean we will bomb anything. But no one should think that Israel has come to terms with terrorism. We will fight back. How, where and in what way, we will decide."

Correspondent Jenkins added from Beirut:

A few tense moments occurred today despite the increasing prospects of peace. After a night of sporadic small-arms and occasional rocket firings in and around the area of the Lebanese parliament, there were at least two car-bomb explosions in West Beirut.

The blasts prompted guerrillas in the streets to fire submachine guns in the air and kept nerves tense in a capital that is seeking to return to normal. Shops are opening and traffic increasingly jams thoroughfares.

This evening, another car bomb was discovered in the parking lot behind the Lebanese Ministry of Information. Police said a young woman, who they identified as Nila Farah, was seen parking the car and leaving hurriedly. They said she drove to the site from East Beirut.

The woman was arrested and police said they found 220 pounds of TNT with a timed trigger. The bomb was dismantled by demolition experts, who said parts of it bore Hebrew lettering. They speculated that the explosives had originated in Israel.