Lowe finished with ease after being put through on goal by Chris O'Grady on 50 minutes.
In a low-key first-half, Wednesday's Chris Lines sent a free-kick straight at Shots keeper Jamie Young and Lowe fired wide from a good position inside the area.
Aldershot's Mark Rankine saw his angled effort easily saved by Richard O'Donnell, while Wednesday skipper Rob Jones charged down a shot from Luke Guttridge.
After the goal, Rankine and Danny Hylton both headed off-target as Aldershot threatened to equalise.
Lowe should have made the game safe for Wednesday when he was through on goal, but he missed the target, while O'Donnell came to his side's rescue with a good save to keep out Adam Smith's firm shot.
O'Grady twice threatened the visitors' goal late on, with one effort saved and another deflected off-target, while Jermaine Johnson shot wide following a good run as Wednesday sealed their place in the draw for the next round.