The winner came when second-half substitute Hassan Sulaiman tussled with Fleet substitute defender Scott Gooding, winning a Shots free-kick 25 yards out. With Fleet holding on to a point at that stage, Darren Barnard rifled in a cross from the left for Sills to swoop in for the winner.
It was a cruel blow for Fleet who, having lost Lee Protheroe on the half-hour mark, had clawed their way back through Charlie MacDonald's second strike equalising for Fleet with only four minutes left. But a Brian Tinnion-inspired Aldershot - making his debut - emerged with all three points, pushing the Shots further away from the drop zone.
Aldershot took the lead after five minutes in a lightning quick move down the right. Matt Somner beat Fleet's Bobby Bowry and sent over a pin-point cross which found Sills out- jumping the Gravesend defence to powerfully head in the opener.
Fleet were level in controversial circumstances with their first real attack Matt Somner was adjudged to have handled in the area and Macdonald clinically despatched the resulting penalty, sending Nikki Bull the wrong way.
The Shots restored their lead on the half-hour mark through another penalty. Debutant, and former Bristol City player- manager, Tinnion's corner was met by Somner, whose header was sailing into the net until former Shots player Protheroe handled on the line. He was shown a straight red, with Darren Barnard burying the penalty, giving Craig Holloway no chance.
Gravesend drew level through MacDonald's second of the afternoon, lobbing Bull from close in after some fine work by substitute Luke Moore. The draw seemed to be a fair result until Sills swung it for the Shots.