Saudi Arabia today publicly backed Syria in the Middle East missile crisis but sent a high-level emissary to Damascus amid speculation that the Saudis were counseling restraint and seeking compromise to head off a new Arab-Israeli war.

The Saudi approach was formulated as special U.S. envoy Philip C. Habib conferred again with Saudi officials and spent his second night in Riyadh on his peace mission, now 11 days old. It came amid these related developments:

The Israeli Cabinet agreed unanimously to give the United States more time to try to find a diplomatic solution to the missile crisis although Prime Minister Menachem Begin said Israel still insisted that the missiles be removed from Lebanon.

Syria forcefully denied it had ever made a secret agreement with Israel about where Syrian troops could go in Lebanon and what weapons they could use.

Lebanese Christian military commander Beshir Gemayel, who was held at arm's length by previous U.S. presidents, left Beirut for high-level consultations with the Reagan administration.

Defense Secretary Caspar W. Weinberger cautioned that the Syrian-Israeli crisis still contained "a great deal of danger" but said that as long as the various sides allow Habib to continue negotiations, "there is still substantial hope."

A Saudi Foreign Ministry statement today said that Syria, which has had more than 200,000 troops in Lebanon since 1976 as an Arab League peace-keeping force, "has the national task of protecting the independence and territorial integrity of Lebanon in the face of Israeli aggression." The statement noted that the Lebanese government retains formal command jurisdiction over the Syrian force there.

Damascus Radio quoted Saudi Foreign Minister Prince Saud Faisal as pledging his country's "full support" for Syria "in the face of Israeli provocations and challenges."

The various Saudi statements, however, left unclear whether that country has agreed to resume payments to Syria to support the force in Lebanon. Saudi Arabia and Kuwait, the chief sources of the $15 million a month needed to maintain the force, reportedly cut off funding earlier this year in a disagreement with Syria over accountability for the expenditures.

Sheik Abdul Aziz Tuweijeri, deputy commander of the Saudi National Guard, was sent to Damascus today with a letter to Syrian President Hafez Assad from King Khalid.

The contents of the letter were not disclosed but earlier, Israeli and Lebanese leftist newspaper suggested Habib would urge a resumption of the Saudi payments in exchange for Syria's removal of the ground-to-air missiles in eastern Lebanon.

The antiaircraft missiles, were brought into Lebanon April 29, the day after Israeli warplanes shot down two Syrian helicopters in Lebanon. Israel charged that Syria had been using the helicopters to attack Lebanese Christian forces under Syrian siege in a mountain stronghold east of Beirut.

While the fighting in the mountains has waned in recent days, there was a surge of violence in Beirut today with the worst artillery bombardments in a week. Hospital sources reported at least 20 dead in the shelling, with Christian east Beirut mostly on the receiving end of Moslem and Lebanese leftist gunfire.

In Jerusalem, the Israeli Cabinet set no time limit for Habib to resolve the issue of the missiles, Washington Post correspondent William Claiborne reported.Instead they decided to wait until Habib completes his travels between Saudi Arabia and Syria and goes to Israel, probably Monday or Tuesday.

"There is no time limit whatsoever," Begin said after the meeting. "We just want to be able to continue the diplomatic moves until they are exhausted. We don't want war. The question is, how to restore the status quo ante in Lebanon, on which the United States and Israel are in full agreement."

Deputy Israeli Defense Minister Mordechai Aippori, emerging from the same meeting, said, "As long as there is a flicker of hope along the diplomatic route we will exhaust that pursuit. The flicker still exists."

The Syrian government, in a statement broadcast by Damascus Radio, meanwhile, denied suggestions that it had tacitly accepted a U.S.-devised agreement with Israel in 1976 by which its forces would not enter southern Lebanon and it would not deploy antiaircraft missiles in Lebanon, Washington Post correspondent Stuart Auerback reported from Damascus.

Joseph J. Sisco, former U.S. under secretary of state, said Friday in New York that he helped draft such an agreement in 1976 in an effort to prevent outbreaks of hostility between Israel and Syria, but he did not say whether the agreement had been accepted by either side. Syria has long contended that there was no such agreement.

"No agreement, direct of indirect, tacit or explicit, has even been reached between Syria and the Israeli enemy," the Syrian statement said today.

Diplomatic observers noted, however, that even if it had tacitly entered into such an agreement, Syria would be unlikely to acknowledge it publicly.

The Reagan administration's growing interest in the Lebanese Christians and especially in Gemayel, who heads the Israeli-backed militia of one of the major Christian factions, became apparent 10 days ago when Habib, at the start of his mission, spent two hours with Gemayel.

Beshir, 35-year-old military leader of the Phalange Party, headed by his father, Pierre Gemayel, is expected to see national security adviser Richard Allen and Nichols Veliotes, assistant secretary of state for Near Eastern affairs, as well as Sen. Jesse Helms (R-N.C.) and other influential conservative members of Congress during his visit to Washington.

Reflecting Gemayel's new aura of respectability and weight abroad, Karim Pakradouni, one of his principal advisers, said in an interview with Beirut's Monday Morning that the missile crisis has made Christian militias a "principal element" in any regional settlement. The militias, he said, have "become a factor that some, like the Soviets, consider important and that others, like the United States, look upon favorably."