Aldershot missed a glorious opportunity to take the lead after 13 minutes when Jeff Goulding blasted over from six yards following Peter Vincenti's flick-on.
After 19 minutes a cross from Wycombe's Matt McClure found Dean Morgan, whose header came back off the inside of the post into the arms of Aldershot goalkeeper Jamie Young.
Wycombe came close again in the 35th minute when Sam Wood's scrambled effort was headed off the line by Terrell Forbes.
The Shots finished the first-half brightly and forced Wycombe keeper Matt Ingram into his first save after Sonny Bradley pounced on a mistake by Gary Doherty.
Five minutes into the second-half Wycombe took the lead with McClure taking advantage of a defensive slip to score his 11th goal of the season.
Aldershot pressed for an equaliser and almost got it five minutes later, Ingram pulling off a superb point-blank save from Paul McCallum.
The visitors duly levelled after 68 minutes, Craig Stanley drilling the ball into the bottom corner from 20 yards after good work by Goulding.
But the visiting fans' joy was short lived as with 15 minutes to go McClure put through Grant who coolly finished against his former club for his 11th of the season.