Chester succumbed to defeat in Greater London thanks to goals either side of half time by Welling.
Reece Harris and Ricky Wellard were the scorers - before Ross Hannah restored late hope in added time; as the Blues slipped to 15th in the table.
John Rooney had the first opportunity of the game - forcing King into a good save with a long range effort.
Following an impressive 1-0 win against Eastleigh on Saturday in which Steve Burr's side created numerous chances - it was more of the same at Park View Road in the first half.
George Thomson headed over from close range, Craig Hobson headed straight at King - and Rooney hit a long range drive just wide.
The hosts were always a threat through the dangerous Ricky Wellard - he forced Jon Worsnop into a smart stop before firing an acrobatic effort just past the post.
Chester's biggest chance of the half came from Hannah - he latched onto a Higgins cross with a bicycle kick which struck the post.
Nevertheless, the missed chances would prove costly as Harris gave the Wings the lead - a succinct lob over Worsnop on the counter attack.
The home side would see a Corne effort hooked off the line by Johnny Hunt - before Hobson smashed a shot wide on the turn at the other end.
But the sucker punch arrived on 70 minutes - when the influential Wellard prodded home following a goalmouth scramble.
Steve Burr's side struggled to conjure up any real chances at a comeback; Jordan Chapell denied a free run on goal by a superb Jefford challenge.
Two subs would go close late on - Kane Richards striking the post before Sam Hughes forced King into another save on the rebound.
Hannah stabbed home following a Chapell corner in added time - but it was too little, too late.
Welling: King, Jefford, Chambers, Lee, Lokko, Nortey, Porter, Wellard, Corne, Harris, Vidal (Kabba 81).
Subs Not Used: McEntegart, Obafemi, Taylor, Awotwi.
Chester: Worsnop, Higgins, Heneghan, Sharps, Hunt, Shaw, Rooney (Hughes 72), Thomson (Mahon 65), Chapell, Hannah, Hobson (Richards 66).
Subs Not Used: Forth, Kay.
Scorers: Harris 43, Wellard 70; Hannah 90+3.
Referee: Robert Whitton.