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Welling suffered a fifth successive National League defeat away from home on Tuesday night.
A deflected Richard Brodie goal four minutes into the second half was enough to settle the contest.
Welling finished the game with 10 men after Barney Williams collected a second caution eight minutes from time.
The hosts hardly had to get out of second gear after the break as, once behind, Welling never looked like getting back into the game.
Three bookings inside the first 22 minutes hardly reflected a game where the only physical challenge of note came from former Welling midfielder Jake Gallagher on the stroke of halft-time.
Gallagher went in two-footed on George Porter, clearly winning the ball, but it was a challenge that would have surely been punished had it come last season while in a Wings shirt.
Williams was cautioned for a late foul for Welling earlier, either side of bookings for Aldershot's Jim Stevenson and right-back Cheye Alexander.
Porter was Welling's brightest attacking threat. Keen to receive the ball and run at the Aldershot defence, he lacked support at times but still provided enough danger for the visitors.
The best opening of the half went to Aldershot but Stevenson's close-range attempt went over following good work by Sam Hatton down the right wing.
Welling had plenty of openings but barely tested keeper Dan Thomas despite dominating possession, particularly in midfield.
Once ahead, Aldershot always looked in control and Charlie Walker went close to a second goal when he fired across the face of goal midway through the second half.
They played out time with relative ease against a Welling side who are without a win since the opening day of the season.
Welling: McEntegart, Jefford, Chambers, Porter, Lee (Nortey 88mins), Obafemi (Adeyinka 54mins), Lokko, Corne, Harris, Williams, Vidal (Wellard 68mins). Subs not used: King, Taylor.
Attendance: 1,321.