Anderson's name now will be submitted to Gov. Gary Herbert for special appointment to the seat, which represents Beaver, Garfield, Iron, Kane, Millard and part of Washington counties.
Also vying for the seat were Millard County GOP Chairman Peter Greathouse, Keith Harrison, of Parowan, and Allen G. Smith, of Delta.
Each candidate was allowed to speak to delegates for five minutes before voting began.
Anderson said in the past 20 years, government spending has increased 261 percent, while the population has increased 62 percent. He also said he thinks the Legislature devotes too much time to special interests, citing House Bill 477 as an example.
The measure would have made changes to the state's open-records laws to exempt text messages from public scrutiny and increase the cost of records requests.
Anderson said the measure never should have been introduced. In late March, after a public outcry, lawmakers repealed it.
"Transparency, in reality, is still the rule," he said.
State GOP Chairman Thomas Wright opened the meeting by asking for a moment of silence to honor Stowell, who died April 17 at his Parowan home after a battle with cancer.
"I know it gives us no pleasure to be here tonight to hold his replacement," Wright said. "He'll hold a dear place in our heart."
Marilee Stowell, who was emotional at times during her speech to delegates, said it was a tender moment for her.
"We have been on a tough journey, and the outpouring of love and friendship to us has just been overwhelming," she said.