Wrexham scored the opening goal after just 11 minutes, as Palmer fired home after a cross was parried away by Yeovil goalkeeper Joe Day. The Red Dragons then had to deal with an injury scare at the next stoppage, as Aaron Hayden limped off and was replaced by James Jones.
Yeovil almost had a stunning equaliser on the stroke of half-time, as Jordan Young sent a 30-yard free-kick against the post, with Mark Howard beaten. Instead, Wrexham added a second before the interval, as Andy Cannon slammed a half-volley at goal, and it squirmed under the outstretched hand of goalkeeper Day.
The Red Dragons remained firmly in control for the second half, and comfortably secured their progress to the fourth round of the FA Cup, as Sam Dalby added a third in stoppage time.