U.S. Marines today extended their presence in the Lebanese capital to the Christian eastern sector, sending their first patrol into one of the most devastated areas along the old "Green Line" that for seven years divided this war-battered city into sectarian parts.

The new Marine mission, carried out in conjunction with the Lebanese Army and French and Italian troops, began without mishap after a one-day delay due to a minor disagreement among the three Western members of the multinational peace-keeping force over how the patrols were to be conducted.

First Lt. Christopher O'Connor, who led one section of the 14-man patrol, said later it had gone "exactly as planned." He reported after touring the eastern sector that the Lebanese had responded "very positively" with smiles, waves and shouts of "hello."

At one point, he said, a civilian had yelled to him, "Give my regards to President Reagan."

Reporters following the convoy of four jeeps with the American flag flying from radio antennas saw many Lebanese, particularly children, stopping to greet the Marines but noted no great outpouring of emotions either for or against the U.S. presence.

With French and Italian patrols a common sight throughout most of the capital, the appearance of the new American patrols was hardly a novelty for most Lebanese, who have endured foreign troops on their soil for many years.

Even with the new daily duty of patrolling the eastern sector, the number of Marines exposed to risk will be limited. Each patrol consists of 12 Marines accompanied by one Lebanese Army liaison officer and an American interpreter and lasts for only three hours.

The Marines carry M16 rifles and 45-caliber pistols while two of the jeeps are mounted with 60-mm machine guns. But the guns are not loaded, although the soldiers carry clips of ammunition with them.

The Marines are under instructions to avoid all confrontations with either the Christian Lebanese Forces militia, which has long controlled the eastern sector and is still present there in small numbers, or any Israeli troops they may see while on patrol.

O'Connor, from Metuchen, N.J., said their orders were simply to report any sightings of armed militiamen or Israeli soldiers to Marine headquarters. If fired on by anyone, he said, the Marines are to "immediately break contact" and ask for assistance from the Lebanese Army.

"No member of our patrol today saw either Phalangist or Israeli troops," he said. "At no time did we observe anything that could be construed as life-threatening or out of the ordinary."

The Christian militiamen are known locally as "Phalangists" after the political party to which most of them belong.

The Marine patrol today went along a 4 1/2-half-mile route through Ain Remmaneh and Furn ash Shabback, two mixed Moslem and Christian suburbs on the southeastern edge of the capital where the 1975-76 civil war erupted and fighting was the most fierce.

The American, French and Italians have finally agreed on a system of rotating their patrols in three-hour shifts along three different itineraries, which are being kept secret, along with the exact schedules, for security reasons.

French and Italian patrols began Wednesday but the American ones were delayed a day reportedy because orders did not come on time from Washington.

The main purpose of the patrols, according to Marine spokesman Capt. Dale Dye, is "to extend the MNF multinational force presence while the Lebanese armed forces and government go to work on establishing control over the whole Beirut area."

President Amin Gemayel had come under growing pressure from Moslem leaders in West Beirut, where the Army established control last month, to demonstrate government impartiality and move the Army into the Christian sector as well. His present effort to get parliament to approve a new 10-member Cabinet and his request for sweeping emergency powers to rule by decree for the next eight months made this all the more imperative.

As it is, the Lebanese Army has had to agree to forego a house-by-house search of homes and shops in the Christian sector and to allow the Lebanese Forces militia to remain intact there, albeit in its barracks and without arms on the streets.

Special U.S. envoy Morris Draper met with Syrian Foreign Minister Abdel Halim Khaddam in Damascus Thursday during a brief visit to the Syrian capital, Reuter reported. The talks centered on the withdrawal of foreign troops from Lebanon. Draper was reported by Agence France-Presse to have returned to Lebanon later in the day and met with Foreign Minister Elie Salem.