Jon Nwannunu came off the bench to score 11 points for Asheville, winners of seven consecutive home games.
Winthrop (10-12, 4-6 Big South Conference) built a 26-18 first-half lead but Asheville outscored the Eagles 9-2 over the final 3:22 to trail 28-27 at the break.
The Bulldogs continued their run in the second half, outscoring the Eagles 23-13 over the first 10 minutes to pull away. Winthrop got no closer than six the rest of the way and Asheville took its largest lead, 67-53, on a Hornsby 3-pointer.
Asheville (14-10, 8-2) shot just 38.3 percent (18 of 47) but made 28 of 32 shots from the stripe. Hornsby and Atkinson combined to make 18 of those free throws.
Derrick Henry led Winthrop with 15 points.
Authorities say former Michigan State basketball player Keith Appling has been charged with carrying a concealed weapon and other crimes.
Point guard Jaylen Fisher, who reopened his recruitment after coaching changes at UNLV, has committed to TCU over a long list of schools.
Forward Kennedy Meeks, who has averaged 9.4 points and 6.5 rebounds at North Carolina, has pulled out of the NBA draft and will return to the Tar Heels.