And Sharp needed no second invitation to pounce with a goal that changed the game, as Chesterfield had bossed the first half and could have been in front.
Striker Colin Larkin should have done better when he dragged his shot wide after breaking through the middle, while skipper Mark Allott had a stinging volley well blocked by goalkeeper Joe Murphy.
Striker Caleb Folan flashed a header just wide of the far post and winger Paul Hall should have done much better with a totally free header in front of goal.
But United's biggest scare came in the 31st minute when they failed to clear a corner and saw a shot from Allott cleared off the line by Marcus Williams.
Scunthorpe had offered little threat - until Sharp's early second half goal.
Then the home side finally found some form and Roche had to keep his side in the game with a string of fine saves.
After Sharp had squandered a good opportunity to make it 2-0, Roche went full length to turn away two cracking drives from Scunthorpe striker Andy Keogh. And he produced another great save to keep out a close range volley from Ian Morris.
Chesterfield brought on Wayne Allison in the closing stages as the searched for an equaliser, but the home defence closed ranks.