Ian Thomas-Moore and Terry Gornell struck either side of half-time for Rovers at the Recreation Ground.
Marvin Morgan came close for the hosts after 19 minutes, twisting in the box only to find the side-netting.
And the hosts really should have taken the lead when Scott Donnelly’s free-kick landed at the feet of Anthony Charles, but with the goal at his mercy he could only divert the ball into the path of grateful keeper Luke Daniels.
Just before the interval Tranmere took the lead against the run of play, Thomas-Moore volleying past Mikhael Jaimez-Ruiz from an angle after his original cross was blocked by Charles.
Moments after the restart the visitors doubled their advantage, Gornell collecting Shaleum Logan’s deep cross and squeezing home from an acute angle.
Aldershot pressed for the remainder of the contest but were unable to force extra-time, Oliver Bozanic’s scrambled finish in injury time a mere consolation.