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Welling bounced back from last weekend's FA Cup exit with a spirited display against leaders Barnet - only to be beaten in injury time.
The Wings looked on course for at least a point against the Vanarama Conference pacesetters but John Akinde fired home in the 91st minute to hand Martin Allen's men maximum points.
It was particularly cruel on Welling who grew in stature as the game went on and could have easily been ahead by that stage themselves after three great openings in the closing stages went begging.
Barnet dominated the early exchanges and had plenty of possession, but they failed to seriously text keeper Benji Buchel, who kept his place with Jonathan Henly left on the bench by boss Jamie Day, who himself returned after suspension.
The visitors went ahead on 25 minutes when Lee Cook's corner fell to the feet of John Akinde, who fired home from 10 yards.
Akinde then headed over Mauro Vihete's centre four minutes later before Welling started to push forward themselves before half-time.
Harry Beautyman volleyed straight at keeper Graham Stack but the former Arsenal stopper was beaten two minutes before the break when he failed to hold on to Chris Bush's free-kick and Joe Healy bundled home the loose ball.
An even second half saw little between the two sides but Welling had three glorious chances as Jake Gallagher, Beautyman and Keiron St.Aimie all failed to convert.
Barnet then delivered the ultimate blow when a long ball sent Akinde clear and he finished despite the best efforts of Buchel.