The council had voted 2-to-2 Friday in an effort to choose between Chavez and Councilman Kirk Mascaro, KSL-TV, Salt Lake City, reported Monday.
The tie was broken and Chavez was sworn in after his name was drawn from a hat by City Attorney Gene Strate.
Chavez had retired two weeks ago from the Carbon County Sheriff's Office after 25 years and previously served on the Wellington City Council for 15 years.
"Kirk [Mascaro] and I grew up together," Chavez said, "and I'm pretty sure I don't think there's going to be any animosity between us. We'll work together just fine."
Chavez will serve until January 2014, which would have marked the end of former Mayor Dean Armstrong's term in office. Armstrong resigned Feb. 23, two days after his arrest for a suspected third driving-under-the-influence offense.