A makeshift home team missing eight regulars proved more than a match for the promotion-chasing Shots until Crane's dismissal for handling outside his area in the 30th minute.
Burton stunned the visitors by taking the lead in the 15th minute when Chris Hall's cross from the left found its way across the area for Laurie Wilson to lash a superb right-foot shot into the top corner of the net.
Andrew Corbett spurned a chance to put Burton further in front as the visitors struggled to find their composure in a shaky start.
But Tim Sills settled their nerves in the 22nd minute when he latched onto a throughball from Jake Robinson and rode Crane's challenge before slotting home from a narrow angle.
Crane's dismissal proved the pivotal moment, as the referee harshly adjudged the keeper had handled after he appeared to chest the ball down.
Worse still for the home side, substitute keeper James Coates' first task was to retrieve the ball from the net after being beaten by Darren Barnard's superb 25-yard free-kick.
The Shots went further ahead in the 43rd minute when Sills rose at the far post to head back a David Lee cross for substitute Brett Johnson to nod into the empty net.
The visitors went into cruise control in the second half and were denied further goals by the heroics of Coates.
He saved superbly with his legs to deny Gary Holloway in the 58th minute and tipped a blistering volley from the same player over the bar five minutes later.
Late in the game Coates produced more outstanding saves, this time to deny Jon Challinor and substitute Nick Crittenden.