The Senate Foreign Relations and House Foreign Affairs committees yesterday approved legislation prohibiting the sale of $354 million in arms to Saudi Arabia, marking the start of the administration's uphill struggle to gain congressional approval of the controversial proposal.

The Senate panel voted 11 to 6 in favor of a resolution introduced by Sen. Alan Cranston (D-Calif.), blocking the sale, while the committee in the House approved a similar resolution introduced by Rep. Mel Levine (D-Calif.) with 222 cosponsors.

Cranston already has the apparent support of 62 other senators, which appears to guarantee that a large majority in the Senate as well as in the House will vote initially against the proposed sale of 2,600 air-to-air and ground-to-air missiles and other munitions.

President Reagan is expected to veto the prohibiting resolution in a bid to uphold the sale. But Congress could then override his veto, if two-thirds of the members in both houses vote to do so.

Currently, opponents of the Saudi arms sale say they are still short of the necessary votes -- four in the Senate and 66 in the House -- to override a veto. Administration supporters predict that some House and Senate members who vote against the arms sale when the resolution comes to the floor, perhaps next week, will switch their votes when it comes to taking on the president.

Levine, who is leading the House opposition to the sale, said yesterday he is optimistic that he will find the additional override votes.

In a separate matter, the House Foreign Affairs Committee yesterday voted 22 to 18 in favor of a bill introduced by Rep. Lee H. Hamilton (D-Ind.), chairman of the House Permanent Select Committee on Intelligence, that would end further U.S. covert aid for rebels fighting against the Marxist government in Angola. The administration has already begun supplying anticommunist guerrilla forces there with $15 million in military assistance, including Stinger antiaircraft missiles.

The Hamilton bill, which now goes to the House floor, would require congressional approval of any U.S. military aid or paramilitary operations in Angola, thus making it overt rather than covert as it is now.

In an effort to keep alive the Saudi arms sale, the administration on Tuesday began its own high-level lobbying campaign in Congress. In letters to Hamilton and House Foreign Affairs Committee Chairman Dante B. Fascell (D-Fla.), Secretary of State George P. Shultz warned that failure to approve the Saudi arms sales would "greatly damage our national interest."

Shultz said the sale was vital to send a "clear political signal" to to friends and adversaries in the Middle East, and to reaffirm "our reliability and credibility as a security partner" of the Saudi kingdom.

Shultz met yesterday in a closed door session with the entire Senate to urge support for the administration's arms proposal and to answer questions on a range of foreign policy issues. It was not clear whether Shultz changed any minds.

Cranston and other opponents have argued that the United States should not sell Saudi Arabia any more arms because of its public support for Libya during the latest confrontation with the United States. The opponents also note Saudi financial aid -- $85 million annually -- to the Palestine Liberation Organization, which they regard as a terrorist group responsible for many recent attacks on U.S. and Israeli interests in the Middle East.