Demonstrating its distaste for the Carter administration's policy in southern Africa, the Senate voted overwhelmingly last night for the speedy removal of economic sanctions against Rhodesia.

By a vote of 75 to 19, the Senate adopted a "sense of the Congress" resolution which declared that the recent Rhodesian elections were free, that the new multiracial government in Salisbury satisfied U.S. demands for black majority rule and that sanctions therefore should be lifted.

The vote was a blow to the administration. But, as senior officials pointed out later, it could have been a little worse. At least the Senate did not vote for legislation that would have legally bound the United States to lift sanctions at once.

But this distinction essentially was cosmetic, since the Senate so decisively went on record as favoring U.S acceptance of the "internal solution" in Rhodesia worked out by Ian Smith and other white leaders there.

Sen. Jesse A. Helms (R-N.C.), the leader of Senate supporters for the internal settlement, declared himself delighted with the vote. Helms said the Senate had sent a message to President Carter that "enough is enough" and that it was time to change the administration's approach to rhodesia.

Aides to Helms said it was clear that the Senate eventually will vote to compel the administration to lift the sanctions.

The legislature language on Rhodesia was tacked onto the annual State Department authorization bill, which the Senate later adopted, 84 to 10. The bill includes a provision that would make appointees to the posts of national security adviser to the president and deputy adviser subject to Senate confirmation.

After finding that the Rhodesian elections were free and would lead to legitimate black majority rule, the Senate called on President Carter to report to Congress by June 30, at the lates, on whether he found the Rhodesian voting to be "free and fair."

Under legislation approved by Congress last year, if Carter were to find that the elections did meet that standard, he must lift the sanctions.

If carter finds that the elections were not free and fair Congress still could mandate an end to the sanctions. Such action would be subject to presidential veto, but even Carter's allies in the Senate seemed gloomy about the ultimate prospects for maintaining the administration's firm line on Rhodesia.

The first public statements by officials of the new Bristish government have reinforced the view that the previous Anglo-American policy is likely to be revised substantially soon.

The old British Labor government had been working with the Carter administration to withhold recognition for the Salisbury government until it negotiated a Settlement with black nationalists in the Rhodesian Patriotic Front who have been waging civil war against the Smith regime. This policy was policy with black Africans, but has become increasingly unpopular in Congress.

A senior official involved in the making of administration policy on Africa tried to put a hopeful light on the Senate vote, saying it at least gave the administration time "to work out an orderly approach to the sanctions question and to the larger problem of finding a solution to the Rhodesian problem."

But this official and others from the administration who were in the Capitol for yesterday's vote were clearly chagrined by the result.

The administration tactic had been to head off Helms yesterday by winning floor approval for legislation requiring the president to make an early report on the Rhodesian elections. Helms wanted the Senate to order a lifting of sanctions without waiting for a presidential report.

Helms' effective parliamentary maneuvering made it impossible for the majority leader, Robert C. Byrd (D-W.Va.), to get a clean up-or-down vote on the legislation the administration approved. Instead, Byrd had to permit the Senate to vote first on the "sense of Congress" resolution that endorsed the Rhodesian elections and recommended an end to sanctions.

Helms happily endorsed this resolution, introduced by Richard S. Schweiker (R-Pa.) and Dennis DeConcini (D-Ariz.), and that is what carried 75 to 19.

Sen. Paul E. Tsongas (D-Mass.), one of the few senators to raise his voice against this vote on the grounds that sanctions should not be lifted, said the Senate's decision "will come back to haunt us."

Tsongas said the vote was a misleading signal to black Africa because it exaggeratedd Senate sentiment for lifting sanctions. Other senators denied that the vote was misleading.

Rep. Stephen J. Solarz, (D-N.Y.), chairman of the House Committee on Africa and a staunch supporter of administration policy, said the Senate vote was "premature and unfortunate."

I the absence of presidential leadership and a willingness to fight for his [Carter's] policy, the changes of maintaining that policy are dim," Solarz said.