But Syria said it will run for a seat on the Geneva-based Human Rights Council in 2013, when Kuwait was supposed to be a candidate.
The Kuwaiti and Syrian ambassadors announced the swap after the 53-member Asian Group met behind closed doors and endorsed the deal.
Syria's U.N. Ambassador Bashar Ja'afari told reporters his government's decision to delay its candidacy had nothing to do with the current unrest in the country and was not the result of pressure.
"It's a sovereign decision based on the Syrian government's will to reschedule the timing of our candidacy ... based on reconsidering our priorities on the list of U.N. candidacies," he said.
In a copy of his speech to the Asian Group meeting, seen by The Associated Press, Ja'afari said the delay was also taken "in light of the number of reform measures that the government began to implement lately in all fields."
Syria was one of four candidates selected to fill four open Asian seats on the Human Rights Council in the May 20 secret-ballot election in the U.N. General Assembly.
It was considered certain to win unless another candidate entered the race or it failed to win a majority of votes in the 192-member General Assembly.
Rights groups and a growing number of governments had waged a behind-the-scenes campaign to prevent Syria from being elected to the Human Rights Council after the Asian Group announced its slate in January, which also includes India, Indonesia and the Philippines. The campaign intensified after the government's attempts to crush an uprising challenging the Assad family's 40-year rule that began in March.
U.N. diplomats said Mongolia had considered entering the race, but with the Kuwait-Syria swap that now appeared unlikely.
U.S. Ambassador Susan Rice welcomed Kuwait's candidacy and praised the "important step" taken by Asian nations who demonstrated their unwillingness to support "a country whose human rights record is deplorable and ... is in the process of killing its own people on the streets, arresting thousands, and terrorizing a population that is seeking to express itself through largely peaceful means."
Human Rights Watch said Syria should now end its violent crackdown on peaceful protests.
Peggy Hicks, the organization's global advocacy director, said "the election had become a referendum on Syria's violent suppression of protests, and Syria withdrew rather than face resounding defeat."
A coalition of 25 human rights groups headed by Geneva-based UN Watch also welcomed Syria's withdrawal.
"The defeat and shaming of Syria's murderous regime in the court of world opinion should be a morale boost for the courageous citizens of that country who continue to brave bullets and beatings to speak out for their universal human rights," UN Watch's Executive Director Hillel Neuer said in a statement.
The 47-member Human Rights Council was created in March 2006 to replace the U.N.'s widely discredited and highly politicized Human Rights Commission. The council, however, has also been widely criticized for failing to change many of the commission's practices, including putting much more emphasis on Israel than on any other country and electing candidates accused of serious human rights violations.
A major problem in the election process is that candidates for the Human Rights Council, and for many other U.N. bodies, are selected by regional groups where there is a lot of internal horse-trading for seats and support. Regional groups often put up uncontested slates to ensure victory for all their candidates.