It took 15 minutes for the first chance to materialise, as Ryan Williams forced home keeper Nikki Bull into a smart save.
Bull was called upon four minutes later to block Julian Alsop's header from an accurate Williams corner and Alsop headed wide from close range soon after.
Alsop went close again ten minutes before the interval, as the experienced striker showed up well on his Rovers debut.
However, with the visitors having had the better chances, the Shots took the lead early in the second half as Dwain Clarke's cross was converted by Tim Sills from close range.
Soon after Clarke forced an excellent save from Steve Perrin, before Alsop again tested Bull in the Shots goal.
On the hour Alsop finally got his reward with a debut goal, heading home Williams' cross from the left.
Williams was to have a hand in what proved to be the winner 16 minutes from time as his ball into the area was half-cleared by Brett Johnson and Mark Beesley drove the rebound past Bull from the edge of the box.
The home side piled on some late pressure, with Nick Crittenden and Sills going close to an equaliser, while in stoppage time both Aaron McLean and Johnson failed to find the net to rescue a point.