The early goal followed a right-wing corner from Ian Baraclough. Richard Hinds headed back at the far post and the lively Sharp knocked the ball into the net from one yard.
Scunthorpe continued to create the better chances and it was not until Yeovil switched to a 4-4-2 formation that they began to come more into the picture.
Their attack was led by 34-year-old Marcus Stewart who was playing his last match of the three-month loan before returning to Bristol City.
City's directors were at the ground, fuelling rumours that Stewart will be back in the January transfer window on a permanent basis.
The experienced Stewart did his best to get his colleagues going but Scunthorpe keeper Joe Murphy - who has now kept four clean sheets in succession - was seldom troubled.
Yeovil started the second half with Leon Best coming on up front for his debut after joining the Somerset club on Thursday on a month's loan from Southampton, but like Stewart, he had little joy against a well-marshalled defence.
The second goal that virtually killed the game came on 62 minutes. Andy Keogh and Cleveland Taylor featured in a bright move down the right and when the latter got the ball into the box, Sharp lived up to his name by beating a clutch of Yeovil defenders to knock it home.
Yeovil threw caution to the wind by bringing Wayne Gray into their attack in the closing stages and he went as close as anyone previously with a speedy run and a shot that cleared the bar.
Lee Morris was also thrown into the attack late on but Yeovil still failed to make a real impression and Scunthorpe fully deserved the three points.