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Former Welling striker Ross Lafayette scored the goal that sent Welling another step closer to relegation in the National League on Saturday.
Lafayette won the spot-kick 14 minutes from time and then stepped up to convert it, making it four defeats in a row for Dean Frost's Welling.
They are now eight points adrift of safety with just 12 games remaining.
Welling started brightly with Luke Wanadio forcing an early stop from keeper Phil Smith.
Aldershot's first opening came on 17 minutes when Marek Rodak had to turn behind Lafayette's attempted chip.
Welling, who handed a debut to Ibrahim Kargbo - who was playing for Ryman League Thamesmead - lost on-loan forward Ade Yusuff to injury before the half-hour.
Reece Harris dragged his shot wide from distance before Rodak made a smart double save at the other end.
Welling probably edged the first half but it was one-way traffic in Aldershot's favour after the break.
Kicking up the slope, they stretched what had been a previously compact Welling defence.
Twice Charlie Walker went close to scoring, first being denied by a Barney Williams tackle and then by the legs of Rodak.
It was a one-man show as Rodak denied former Welling midfielder Jake Gallagher as well.
His resistance was finally broken when referee Rob Whitton decided his challenge of Lafayette was a penalty.
The visitors then managed to hold out against a goalshy Welling side who have now gone four matches without scoring.
Welling: Rodak, Chesmain, Wanadio, Nortey, Ferry, Lokko, Harris, Williams, Yusuff (Kabba 29mins), Kargbo, Adeloye (Daniel 79mins). Subs not used: McEntegart, Gayle, Daniel. Attendance: 704.