Sharp, a threat all afternoon, finished well from a sweeping Iron breakaway in the 52nd minute.
Tricky winger Cleveland Taylor was the architect, sprinting from his own half before squaring the ball to Sharp who controlled and side-footed past Keiren Westwood.
Man-of-the-match Andy Keogh wrapped up the three points with the last kick of the game, lofting the ball in from 45 yards with Westwood stranded out of his goal.
It was a deserved win for the visitors who completely dominated for the whole 90 minutes.
The hosts offered little in attack with their first shot on target coming ten minutes before half-time.
Simon Hackney's pinpoint cross found Kevin Gall unmarked but he waited too long and eventually scuffed the poor shot into the waiting arms of Joe Murphy.
Referee Mike Pike was taken ill at half-time with his assistant Mr Duncan taking his place, but a change of official couldn't save Carlisle from a thoroughly deserved defeat.
Sharp's goal after half-time should have kick-started the Cumbrians but instead it was Scunthorpe who raised their game to another level.
Sharp again went close to his second with an expert spin and shot that flew just wide and he should have wrapped up the points with another chance late on.
A long punt from defence should have been dealt with by Carlisle's Peter Murphy but the lively Sharp bullied him off the ball but could only slide his shot wide with the keeper to beat.
Carlisle substitute Neale McDermott went close with a 25-yard volley that flew just over but an equaliser never looked likely.
With the game deep into injury time Westwood stepped to take a free-kick on the halfway line.
But his kick was easily cleared to Keogh, and the Republic of Ireland Under-21 international took a touch before lifting a perfect shot into the net from just a few yards inside the Carlisle half.