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Welling keeper Lee Butcher was their undoubted hero after a series of excellent second-half saves in their FA Cup first round win on Saturday.
First-half goals from Lee Clarke and Joe Healy put the Wings on the way to victory over fellow Skrill Premier outfit Luton but Butcher stole the show in the final half-hour.
He produced four brilliant saves to ensure the Wings progressed to the second round for the first time since 1989/90 when they were finally beaten by Reading in a third replay having knocked out Gillingham in a replay in the first round.
Both sides had chances in an even opening 20 minutes but Welling took the lead on 27 minutes when Ross Lafayette, back in the starting line-up after injury, crossed for Clarke to bundle the ball home at the near post for his seventh goal of the season. It was Clarke's fourth goal in his last seven games and his second FA Cup goal of the season after he netted the winner at Brislington in the fourth qualifying round.
The lead should have been doubled within sixty seconds but Lafayette, having brushed aside Steve McNulty, somehow fired wide of the near post with just keeper Elliot Justham to beat.
Two minutes later, Luton somehow failed to level when, with Butcher out of position, Luke Guttridge collected Jack Obersteller's misplaced pass only to fire wide of an open goal from the edge of the area.
Welling added a second on the stroke of half-time when Lafayette picked out Healy's perfectly-timed run and he fired low beyond Justham into the bottom corner.
Lafayette was then denied a third goal early in the second half when his goalbound shot was blocked by the legs of his own player and Alex Dyer's fizzing drive was deflected over.
Luton got back into the contest on 57 minutes when a right-wing cross was met by the head of Paul Benson. His header was brilliantly saved by Butcher but Benson reacted quickest to the loose ball to nod in from close range.
With the visiting crowd behind them, Luton threw everything at the Wings but Butcher stood defiant. He was down smartly at his near post to keep out Jonathan Smith's left-foot drive and then excelled in the final five minutes.
Benson's downward header, Alex Lawless's one-on-one and Brett Longden's near-post shot were all kept out by Butcher's heroics.
Substitute Kiernan Hughes-Mason nearly added a third at the other end but it was Butcher who took the applause as the Wings now await the second round draw.