The United Nations Security Council gave its approval last night to an American call for a cease-fire in the fighting in Lebanon after President Carter personally obtained pledges of cooperation for American efforts at the U.N. from Soviet leader Leonid Brezhnev and other world leaders.

Meeting in an emergency session that had been sought by the United States, the 15-member Security Council unanimously endorsed a cease-fire resolution that Arab and Western diplomats predicted would be accepted by the Syrian army, which has raked Christian areas of Beirut with heavy bombardments for a week.

Joining an initiative launched at the beginning of the week by France and Saudi Arabs as the fighting escalated in Beirut, the Carter administration mounted an intense diplomatic campaign over the past 48 hours to press the Syrians to stop those bombardments and to pull back from frontline positions in Beirut.

Carter personally took charge of the U.S. campaign Thursday night by contacting Brezhnev, White House and State Department spokesmen said yesterday. They declined to describe the Russian response, but it was reported both in Washington and by diplomatic sources at the U.N. to have been considered by the White House as constructive.

The Soviet Union is Syria's principal arms supplier, and Carter's message requesting cooperation came as Brezhnev was hosting Syrian President Hafez Assad who received a second message from Carter within 24 hours yesterday shortly before leaving Moscow for home.

The Russians and the Syrians had not appeared receptive to cease-fire suggestions earlier in the week but U.N. diplomatic sources said that Syria had been consulted throughout the day as delegates sought to combine draft cease-fire resolutions drawn up by the United States and by Kuwait the current Arab representative on the council.

The compromise resolution was taken up at a formal council session that began at 7:15 p.m. with prior agreement for all members that the draft would be adopted.

The resoltuion called on "all those involved in hostilities in Lebanon" to stop fighting so internal peace and national reconciliation could be restored based on the preservation of Lebanese unity territorial integrity independence and national sovereignty."

The resolution also called on the warring factions to allow the Lebanese Red Cross to enter the battle areas Red Cross to enter the battle areas and evacuate the wounded. U.N. Secretary General Kurt Waldheim was asked to continue his efforts which included the dispatch this week of a special high-level representative to Lebanon.

But there was no mention in the resolution of the stationing of Lebanese army units in buffer zones between the Syrians and the Christian militia units that provoded the latest round of fighting in the three-year Lebanese conflict by attacking Syrian soldiers last weekend.

France had coupled its original call for a cease-fire to a plan for a Syrian "redeployment" in Beirut that would allow Lebanese President Elias Sarkis to station several thousand newly trained men from the shattered Lebanese army in buffer areas.

The United States, which for two years has supported the Syrian presence in Lebanon as a stabilizing force, eventually endorsed the French redeployment plan in what some diplomatic analysts here saw as a turn in American policy.

The Security Council action is expected to reduce domestic political pressures that had been growing both in France and the United States for a halt to what the State Department yesterday called "the intolerable carnage" in Beirut.

A bipartisan group of 17 U.S. senators led by Richard Stone (D-Fla.) sent Carter a letter yesterday urging him to broaden the U.S. call for a cease-fire and to seek expansion of the U.N. force in southern Lebanon and send it to Beirut as a substitute for the Syrian units. These form the bulk of the Arab peace-keeping force sent to police the 1976 crease-fire that was supposed to end Lebanon's savage civil war.

Carter devoted most of his weekly foreign policy breakfast to the Lebanese crisis. Secretary of State Cyrus Vance flew from New York for that meeting then returned to begin the consultations that worked out the compromise draft resolution and arranged the formal Security Council session.

The president was also in contact yesterday with French President Valery Giscard d'Estaing, who was in his final full day of a brief visit to Brazil. France, the former mandate power in Lebanon, has 1,400 soldiers in the 4,500-man U.N. force and Giscard has come under increasing pressures at home to act or lose credibility.

The Russian response to Carter's message appeared to halt growing questioning at the White House of Moscow's intentions in the Middle East after the Camp David summit agreements between Egypt and Israel had put the Soviet Union Syria and the Palestine Liberation Organization on the sidelines.

The White House declined to say whether Carter's contact with Brezhnev was in writing or by telephone.

Growing U.S. impatience with Lebanese President Elias Sarkis for not asking the Syrians to halt their bombardments and to pull their troops back was evident in comments by senior administration officials over the past two days.