Gabriel Zakuani was next to come close for the home side after 19 minutes, but twice he struck wide from close range.
Southampton continued to dominate in the latter stages of the half, but George Boyd so nearly cut the deficit to 2-1 just before the break. However, his effort was blocked on the line by Chaplow.
The home side might have found a way back into the match moments after the interval, but Lee Tomlin could only find Davis from close range.
Southampton ended any possible hope of a comeback, however, when Sharp grabbed his second of the match after 57 minutes to move the away side into a 3-0 lead.
It might have been four when Dean Hammond struck from close range after 70 minutes, but he missed the target.
Tommy Rowe added a late consolation for the home side with a cool finish five minutes from time, but it did little to dampen Southampton's celebrations.