Brown, former superintendent of the Cave City School District, defeated Goings, a local business owner and former JP of Hardy, with a vote of 2,097 to 1,566.
Brown, who served eight years as a Sharp County JP, will face independent candidate Wayne Long in the November general election.
Brown said the growth of the county requires the need for short and long range plans to give county officials some guidance. The plan should include a comprehensive road plan, he said.
In order to encourage and support growth, the county should complete the natural gas feasibility study currently under way, Brown said. The county should also push the highway commission to complete the four-lane corridor from Little Rock to Ash Flat and the 412 east-west corridor, he said.
Weaver, former drug task force administrator, won the support of the county in 2002 when he ran against Michael Steele.
In the constable races winners were Mike Zeiger District 1, Mary L. Wanley District 3, Michael R. Orosz District 4 and Clint Madison for District 6.