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Donovan Simmonds' late winner handed Dover their first Blue Square Bet South win on Saturday since their FA Cup exit.
The scoreline flattered the visitors after Welling dominated the second half but the hosts failed to find a way through a resolute Dover defence.
The game started at a ferocious pace with Lee Clarke's shot blocked, Adam Birchall being denied by keeper Mitchell Walker and then Joe Healy saw his effort cleared off the line - all in the first six minutes.
Luis Cumbers thought he had given Welling a 31st-minute lead but, having beaten Ollie Schulz for pace, Cumbers saw his shot cleared off the line by visiting skipper Rob Gillman.
It needed another fine Jamie Coyle block moments later to deny Birchall after Lewwis Spence' burst forward and shot.
Spence and on-loan forward Danny Kelly both impressed on their Whites debuts although they both faded after the break.
Indeed, after Spence hit the bar with a 49th minute free-kick, it was Welling who looked more likely to score.
Jack Obersteller's shot high and wide, a feat followed by Andy Pugh while Gillman's header back went over keeper Ross Flitney but wide of the goal.
Cumbers headed narrowly wide from Coyle's cross before player/manager Jamie Day's free-kick was comfortably saved by Flitney.
Dover started to look dangerous on the break and Simmonds (pictured) shot and turned from the edge of the area to give his side the lead, via the inside of the post, with 12 minutes left. Martin Hayes' side then comfortably ran down the clock to gain maximum points.