Clinical finishing was the difference as Stafford created more chances, but they could not find a way through as the visitors moved up to sixth position in the table.
The hosts made a positive start and Robin Gibson fired over from 20 yards, before Town's Ryan Williams struck a free-kick straight at the wall.
Aldershot goalkeeper Nikki Bull did well to deny Guy Madjo before the visitors took advantage to race ahead in the 31st minute.
John Grant found Ryan Williams on the left and he delivered a perfect cross to the far post where Gayle was well-positioned to head home his second goal of the season.
Stafford, who created the better first-half opportunities, had a chance to equalise just before the whistle. Nathan Talbott and Madjo combined to set up Ged Murphy, but he blasted over.
The home side survived a scare in the 55th minute when Grant headed a deep left-wing cross towards goal, but Lee Downes cleared off the line.
Aldershot should have extended their lead just after the hour mark, but Dean Williams produced a fine reaction save to deny Grant and Downes was on hand to clear Ryan Scott's shot from the rebound.
But the large group of Shots fans were soon celebrating a second goal in the 68th minute. The dangerous Louis Soares took the ball down the right and crossed towards Grant, who got free from Wayne Daniel and fired home.
Brave Stafford continued to move forward in search of a goal. Both Madjo and Craig McAughtrie saw shots blocked in a scramble before Neil Grayson glanced a header across the face of the Aldershot goal.
The result was put beyond doubt when Aldershot scored a third in the 88th minute. A slip by Daniel allowed Ryan Williams to latch onto Scott's pass and clinically fire past Dean Williams.