Jimbo Watson hit Tennessee's fourth home run in the ninth, a three-run shot for the final margin.
Walter Sevilla had three of the Vols' 18 hits. Alley, Herrera, Chappell, Nick Crowe and Jordan Czarniecki added two apiece.
Dusty Johnson (6-1), who struck out five and walked none, picked up his team-leading sixth victory as he surrendered six runs on 10 hits in seven innings.
He was touched for two home runs in the first by Grant Rembert, a two-run shot, and Daniel Pruitt as UNC Asheville (19-22) took a 3-1 lead. Josh Coyle then added a two-run single in the fifth, while Rembert drove in his third run with a double in the seventh.
Joey Andrews gave up a Bulldog run on Scott Stotlar's double in the eighth, while Steve Wesley pitched the ninth, yielding the final run on a double play.
Dustin Sode (2-1) took the loss as he was chased in the second after giving up eight runs on six hits while issuing five walks with two strikeouts.
Rembert had four hits for the Bulldogs as Jake McConiga scored twice while going 4-for-4, reaching base safely in all five plate appearances.
The Volunteers take on the Class AA Tennessee Smokies in an exhibition contest tonight at 7:15 at Smokies Park in Sevierville.