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Welling survived a late Wrexham onslaught to move out of the relegation zone for the first time since the start of March on Tuesday night.
The Wings, rocked by the late arrival of three players originally named in their starting line-up, came flying out of the blocks and led 2-0 inside 20 minutes.
Ben Jefford, keeper Jon Henly and skipper Jake Gallagher were all late arrivals, meaning boss Loui Fazakerley drafted in Chris Bush, Zach Fagan and keeper Michael McEntegart, who was playing his first game at this level.
Welling went ahead on seven minutes when Dominic Vose's deep corner was headed in at the far post by the unmarked Sam Berry.
They immediately pressed for a second but Kieron St.Aimie fired over after Ross Lafayette nodded down Bush's deep centre.
McEntegart turned Clarke's low 12th-minute drive behind for a corner before Welling scored again. Impressive work by Lafayette saw him pull the ball back for Vose, who fired high into the roof of the net at keeper Andy Coughlin's near post.
Lafayette then hit the bar from Reece Harris' centre, only to be flagged offside, although Harris crashed into the advertising hoarding and his evening was curtailed by injury.
It was a near faultless first half from the Wings as the likes of Bush and St.Aimie produced some of their best football of the season.
After the break, Welling were limited in their attacking play although St.Aimie fizzed a long-range effort just wide of the upright.
It needed a fine one-handed save to deny Wrexham's Kyle Storer midway through the half while substitute Louis Moult nodded wide of the near post when left unmarked from a corner.
A rare mistake by teenager Sam Corne, who again produced a midfield display beyond his tender years, let Wrexham back into the game with eight minutes left. His loose pass was intercepted and Andy Bishop confidently tucked away the through ball.
That set up a nervy finale for the Wings but McEntegart punched away most things that came into the penalty area and then held on as substitute Rob Evans sent an 89th-minute header goalwards.
Referee Colin Lymer played four minutes of stoppage time but Welling held on with a fine rearguard effort and now head to Alfreton on Saturday knowing victory will guarantee their survival with one game to go.
Welling: McEntegart, Fagan (Gallagher 74mins), Bush, Vose, St.Aimie, Lafayette (Akinde 79mins), Berry, Harris (Jefford 34mins), Corne, Osborne, Chambers. Subs not used: Young, Henly. Attendance: 612.