Israeli forces pushed their drive against Palestinian guerrillas to within 10 miles of Beirut yesterday and fought their first major clashes with Syrian jets and troops in the three-day-old invasion.

The Israeli Army command said its warplanes shot down seven Syrian Mig fighters, three of them over northern Israel, while losing none. Syria admitted losing three Migs and claimed downing seven Israeli planes.

After capturing the port of Sidon, Lebanon's third largest city, Israeli tanks and troops drove north in two columns. One pushed up the coastal road and besieged Damour eight miles south of the Lebanese capital. The other struck east, seized strategic high ground in the interior and then turned north to threaten the key road linking Beirut to the Syrian capital of Damascus.

A Syrian communique said its troops were shelling the Israeli column that was approaching the highway. Israel said it had tangled with the Syrians in tank battles near Jazzin east of Sidon.

Israeli jets and ships continued to bombard Palestinian positions, and major fires were reported in Sidon and the port of Tyre, which fell to the Israelis after 48 hours of heavy fighting. Thousands of refugees continued to pour into Beirut, fleeing the advancing troops.

Israeli forces, estimated at 20,000 or more, have now pushed more than 30 miles into Lebanon from the Israeli frontier. They have thus moved beyond the 25-mile border zone that Israel declared it wanted to clear of all Palestinian guerrillas to halt artillery and rocket attacks on northern Israeli settlements. The invaders apparently believe they must control the area around Beirut itself--40 miles from Israel's northernmost point--to accomplish their objectives.

The escalation in fighting with Syria, which has 23,000 troops in Lebanon, heightened the danger that the conflict could widen into a full-scale Middle East war. The Syrians apparently still have not moved against the Israelis with full force, however, and Israeli Prime Minister Menachem Begin urged Damascus to rein in its troops.

"We do not want to clash with even one Syrian soldier," Begin told the Israeli parliament, the Knesset. The legislature voted 94 to 3 to back Begin's decision to launch the invasion, handing him his most lopsided vote of confidence since he was first elected in 1977.

Begin's position also was strengthened by the relatively low casualty figures announced so far by Israel--25 dead and 96 wounded as of midnight Monday. The Palestine Liberation Organization has not yet released figures on casualties, but Lebanon's ambassador to the United Nations, Ghassan Tueni, was quoted as saying that they ran into the "thousands" and that most were civilians.

The Reagan administration labored to appear evenhanded in the crisis as the Lebanon fighting began to overshadow the president's trip to Europe. In an address to the British Parliament, Reagan said that Israel "should bring its forces home" but added that all parties in Lebanon must stop fighting.

"We must all work to stamp out the scourge of terrorism that in the Middle East makes war an ever-present threat," Reagan said.

Secretary of State Alexander M. Haig Jr. dodged answering how far the United States was willing to go to exercise sanctions to force Israel to withdraw.

"There's been no decision in that area," Haig told reporters, saying the administration's goal was "to do all possible to bring about a termination of the bloodshed."

Some administration sources said Haig may travel to the Middle East within the next two weeks to help negotiate a long-term solution that would preserve Lebanon's integrity and guarantee Israel's security. U.S. special envoy Philip C. Habib met again with Begin but declined to comment to reporters on the talks. At the United Nations, Israel rejected a Security Council demand that its troops withdraw unconditionally, saying a pullback was "inconceivable" without "concrete arrangements" to safeguard Israeli citizens permanently. Later, the United States vetoed a Lebanese-sponsored resolution condemning Israel's refusal to withdraw and threatening Israel with sanctions unless it agreed to pull out immediately.

Lebanon and the PLO had accepted the council's call for a cease-fire.

PLO leader Yasser Arafat accused the U.N. peace-keeping force in Lebanon of allowing the invading Israeli Army to use its bases and roads and of collaborating with Israeli troops. U.N. Secretary General Javier Perez de Cuellar said that the 7,000 soldiers in the force had risked their lives trying to block the Israelis and that Dutch troops had planted obstacles in front of advancing Israeli tanks, damaging one.

Arafat also called on other Arab leaders for help, appealing to "their historic and national responsibilities." Iranian Foreign Minister Ali Akbar Velayata said at the United Nations that his government was prepared to send volunteers to Lebanon to support the PLO and had dispatched a special mission to Syria to discuss the matter.