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An injury-time penalty cost Welling a share of the points at Aldershot, leaving them hovering just two points above the drop zone.
Welling started the game well with Zachari Fagan, Dominic Vose and Reece Harris all forcing home keeper Phil Smith into making a save.
However, against the run of play, Aldershot took the lead four minutes before the break through James Stevenson.
But Welling were soon on level terms, Vose replying just before the break, firing in off the underside of the bar.
Karl Duguid almost gave Welling the lead in the second half, only to see his 25-yard effort hit the post and go wide.
And with the game heading for a draw, Aldershot won a penalty in injury-time and Ryan Jarvis converted to break Welling hearts.