Frank received 7,074 votes, or 58 percent, while Gooch collected 5,032, or 42 percent, according to unofficial results Thursday night.
She will serve a two-year term, completing the unexpired term of former mayor Rex Lynch, who resigned in January 2011.
It was one of three special elections on Thursday’s ballot in Anderson County. The other two were for Anderson County Commission District 8 and Oak Ridge City Council.
Anderson County Interim Mayor Myron Iwanski won back his old seat in District 8, easily defeating the current commissioner, business executive Buzz Patrick, who was appointed to the seat in March 2011 after Iwanski was appointed interim mayor. Iwanski, who has said he wanted to complete his term on commission, won 1,354 votes, or 72 percent, compared to Patrick’s 521 votes, or 28 percent.
The district includes Woodland, Emory Valley, and Hendrix Creek.
Oak Ridge City Council member Chuck Hope held on to his seat for now, beating challenger Trina Baughn by a 58 percent to 42 percent margin. A business owner, Hope had 2,402 votes, and Baughn, a communications professional, had 1,725.
Hope and Baughn ran for a special three-month term on City Council. Hope was appointed to the seat after former City Council member Tom Hayes resigned in June 2011. The two could face off again in November. The winner of that election will serve a regular four-year term.
In the race for Anderson County property assessor, Republican John K. Alley Jr. beat Democrat Rick W. Marlowe by 60 percent to 40 percent.
In the federal Republican primary, U.S. Rep. Chuck Fleischmann received 49 percent of the vote in Anderson County, compared to 27 percent for Scottie Mayfield and 23 percent for Weston Wamp.
Mary M. Headrick beat Bill Taylor 74 percent to 26 percent in the Democratic primary for the U.S. House seat.
U.S. Sen. Bob Corker received 82 percent of the votes cast in the Senate race in the Republican primary, trouncing his opponents, and Park Overall collected 27 percent in the Democratic primary, compared to 22 percent for her closest competitor, Mark E. Clayton.
The election results, which include votes from all 27 precincts and early and absentee ballots, are posted on the Anderson County Election Commission website.
In Roane County, Tennessee Rep. Julia Hurley was defeated in the state Republican primary by former Roane County commissioner Kent Calfee; engineer David Morgan beat Teresa Kirkham, the incumbent county property assessor; and former Tennessee Rep. Dennis Ferguson was elected county road superintendent.