The State Department, which has sought for 13 years to bring down Rhodesian Prime Minister Ian Smith, grudgingly agreed yesterday to grant the world's ranking rebel-statesman a visa, which will permit Smith to launch an intensive lobbying campaign on Capitol Hill next week.

Acting over fears among some top aides that a warm reception in Congress could bolster Smith's determination to continue to defy United Nations and other international pressure, Secretary of State Cyrus R. Vance personally made the decision on the long-delayed visa application Tuesday afternoon.

The decision - debated for two weeks within the Carter administration as part of a wider-ranging review of U.S. policy toward southern Africa - was announced yesterday by State Department spokesman Thomas Reston, who said Smith's visa would be granted "on an exceptional basis" in the hope that "the visit can contribute to the process of achieving a settlement of the Rhodesian conflict."

Vance has indicated to Smith that the secretary would welcome a meeting during the visit, a senior U.S. official told reporters after the announcement. The decisive factor in granting the visa was reported to be Vance's confidence that discussions with Smith here will help bring Smith closer to negotiations with African guerrillas fighting to take over the country.

But the senior State Department official, who insisted on not being named, denied that the delay in the visa decision had been linked to new U.S. efforts to get full negotiations started involving the guerrillas and Smith's government. Refusal by the Smith government to accept an Anglo-American negotiation proposal continues to be the main block to direct negotiations, the official said.

The decision to grant visas to Smith and to the three African nationalist leader who serve with him on Rhodesia's ruling executive council was immediately hailed by Sen. S. I. Hayakawa (R-Calif.), the leader of a group of 27 senators who last month issued an invitation to the four Rhodesians.

Hayakawa, who Monday called State Department officials "spineless," said yesterday that he was "extremely gratified. I am glad the State Department has decided to open the entire question to a rational debate in which all sides can be heard."

Sen. Jesse A. Helms (R-N-C.), who earlier this week had quietly threatened to block a number of State Department appointments and promotions if the visa request were refused, also praised the decision, which he described as being "about 13 years late."

But the green light for the single-entry, 30-day visas for Smith and his colleagues was also a signal for immediate criticism from many African nations and black American groups that maintain that Smith is committed to maintaining the kind of white supremacy that has enabled about 200,000 white settlers to rule 6 million blacks.

Three U.N. committees dominated by African countries issued statements expressing "concern and distress." Nigeria, Tanzania and Zambia made separate public statements urging Washington to reconsider the move.

Transafrica, a lobbying group composed of prominent black American political leaders, sent President Carter a letter describing the decision as "a grave error" that would be seen by Smith as an endorsement of Smith's plan for an "internal settlement" of the 6-year-old war with the guerrillas. Earlier, the group said it would seek to block the visa action in court.

The U.S. official briefing reporters said that the State Department had not consulted with the "frontline" African states that support the guerrillas and which have been instrumental in the Anglo-American peace effort before making the visa decision. He added that Britain had left the decision entirely up to the United States and had not expressed an opinion on the legality of allowing Smith to enter the country.

The prime minister make Rhodesia a rebel colony in 1965 by unilaterally declaring independence from British rule rather than accept British plans for eventual black majority rule. Since then, Smith has defied economic sanctions imposed by the United Nations and supported by the United States.

In March, as Soviet and Cuban help increased for the Patriotic Front guerrilla forces, Smith brought Africans into his government and promised to hand over power eventually. The Carter administration has come under increasing criticism from American conservatives for insisting that a final settlement has to be negotiated with the guerrillas, and the visa question became an especially volatile one since guerrilla leaders Joshua Nkomo and Robert Mugabe have visited the United States.

Until yesterday, the State Department has complied with a 1968 U.N. Security Council resolution declaring the white settler government illegal and ordering member countries to refuse entry to Rhodesian officials. In granting Smith a visa, the State Department will waive the requirement that he present a travel document, and thus avoid the question of the legality of Rhodesian passports.

A Hayakawa aide said that Smith and the Rev. Ndabaningi Sithole will arrive in the United States Saturday and plan to appear on television interview programs Sunday. They will hold sessions with the 27 senators who invited them and with other members of Congress interested in hearing them next week, and plan a number of television, newspaper and magazine interviews during their one-week stay in Washington and New York.

Bishop Abel Muzorew and Chief Jeremiah Chirau, the other two members of the executive council, will visit the United States later, a government spokesman said in Salisbury.

The announcement by the State Department came in a four-page statement that contained lengthy justifications for the decision, which the statement maintained "does not imply U.S. recognition of or support for the present Rhodesian administration" or its internal settlement proposal. The Carter administration will continue "to observe our responsibilities as a member of the United Nations under the Security Council's resolution on Rhodesia," the statement added.

The senior offical said that a sharp escalation in fighting in recent weeks and the administration's perception that "a tragic disintegration" has begun in Rhodesia triggered a full-scale review of American options in southern Africa, which was under way when Smith's visa application was received on Sept. 20. The visa decision was delayed "while we sorted out where we thought things stood," the official said.