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Winless Welling conceded a fifth penalty in as many games to lose on Saturday.
It meant visitors St Albans inflicted a fourth defeat in the first five games of the season on Mark Goldberg's Welling side in National League South.
Welling started brightly enough and nearly took a fourth-minute lead. Archie Johnson’s left-wing cross found Dan Walker and keeper James Russell produced a stunning one-handed save to turn his shot against the bar.
Harry Crawford went close on the counter attack as the Wings looked in control.
But that dominance faded after half-hour and the visitors got on top. It needed a point-blank save by Wings keeper Chris Lewington to keep out Josh Hill’s header from Lee Chappell’s free-kick.
Mitchell Nelson was then caught napping as Junior Morias got in behind and was denied by Lewington.
But Nelson did not get away with it four minutes from time, needlessly bundling over Shaun Lucien in the box. Morias tucked away the spot-kick.
After the break, St Albans appeared in complete control and it took Welling time to create openings. The woodwork denied Welling again in the 66th minute when Nelson’s header came back off the bar.
Bobby Devyne’s shot was deflected over and Danny Waldren was denied by Russell as Welling had chances without ever being on top.
Welling were heading for the bottom of the table but two goals in injury time by Oxford City denied Concord their first three-point haul of the campaign.
Welling Lewington, Nelson, Brown, Hayles, Francis, Leacock (Sprague 71mins), Devyne, Waldren, Crawford (Slabber 59mins), Walker, Johnson (Hatton 46mins). Subs: Banton, Agu.
Attendance: 483.