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Striker Tyrone Marsh scored twice as Welling picked up three points in their first Vanarama Conference game since Jamie Day quit to take over at Ebbsfleet.
Marsh scored in either half to help the Wings pick up back-to-back league victories for the second time in as many months and move nine points clear of the relegation zone heading into the festive fixtures.
Under the guidance of caretaker managers Jamie Turner and Jake Gallagher, Welling showed plenty of passion and desire and were ultimately rewarded for their endeavour against Andy Scott's Aldershot team.
The visitors did start brightly and keeper Benji Buchel did well to deny former Salisbury frontman Dan Fitchett, whose rebound was then cleared off the line by Jamal Fyfield.
Welling slowly found their feet and Marsh should have done better when he got behind hamstring victim Mark Phillips and tried to go round keeper Phil Smith, who got a hand to clear the danger.
Phillips was quickly replaced but Aldershot's defence was exposed again moments later as Malachi Hudson burst down the left wing and crossed for Marsh, whose close-range shot was straight at the grateful Smith, who still did well to make a crucial save.
Welling finally went ahead in stoppage time when Marsh was played in behind left-back Chris Barker and he cut inside before firing low beyond Smith for his seventh goal of the season.
The Wings doubled their advantage on 53 minutes when an excellent move saw Chris Bush send Hudson free and his cut-back was collected by Kieron St.Aimie, who steadied himself before producing a low finish into the bottom corner.
It was almost three moments later but Joe Healy's spectacular long-range effort was expertly turned away by Smith.
Aldershot pulled one back midway through the half when a free-kick was never properly cleared and after an initial attempt was blocked, skipper Glenn Wilson slammed in the loose ball from close range.
Dan Holman headed straight at Buchel on 72 minutes but, having survived what pressure Aldershot put on them, Welling wrapped the game up nine minutes from time.
A clever flick by Aristide Bassele sent Marsh clear on the left and he opened his body up to curl a right-foot shot beyond Smith and into the far corner.
It could have been more late on but St.Aimie's long-range shot was brilliantly tipped onto the post by Smith.
Welling appointed Grays boss Jody Brown as their new manager on Saturday evening.
Welling: Buchel, Williams, Fagan, Fyfield, Bush, Healy, Corne, Hudson (Penny 80mins), Bassele, St.Aimie, Marsh. Subs not used: Henly, Ouani, Kiassos, Johnson. Attendance: 756.