The Shots had several chances to open the scoring as keeper James Mann denied Steve Watson after neat play from Tim Sills and Jonny Dixon.
Nick Crittenden's effort was deflected just wide, while Mann denied Sills after a defensive error from Ryan Clarke.
At the other end Gareth Williams' header was just off target as Leigh produced little going forward.
Returning loanee Dixon was also denied by the in-form keeper as he shot from an acute angle.
Shots keeper Nikki Bull fumbled under pressure from Chris Simm, but the defence were on hand to clear the danger.
Then on 35 minutes Leigh midfielder Mark Drew was given his marching orders when a yellow would have sufficed for a challenge on Hugh McAuley.
The opening goal finally arrived in the last minute of injury time in a half when Ty Smith's curling free-kick found the back of the net.
The Shots doubled their lead just four minutes into the second half when Mann grabbed hold of Sills' ankle after an exquisite throughball from McAuley. Mr Desmond had no hesitation in pointing to the spot from where Dixon made no mistake.
The Shots then rained attacks on Leigh's goal as Dixon was denied by the keeper.
Crittendens long-range effort was superbly pushed over the bar by Mann for a corner. Dwain Clarke's delivery was then headed against the crossbar and over by Dixon.
Clarke himself was just wide although it looked like the keeper just got a touch on it, while substitute Chukki Eribenne was also denied by the keeper from close range.
However Aldershot had done enough to hold on to the valuable points.