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Two goals in the last three minutes handed Welling back to back wins for the first time this season on Saturday.
The Wings dominated from start to finish against a Weston side who lacked any attacking intent.
But they nearly threw away the three points in an action-packed finale to the Blue Square South game.
It took Welling 73 minutes to make the breakthrough when substitute Adrian Stone's cross fell for Michael Deane and his deflected shot fell perfectly for Tommy Whitnall to score his second goal in as many games.
That looked enough for the Wings but Weston equalised with 10 minutes left when slack defending saw a deep cross fall for Jack Compton, who produced an impressive finish at the far post.
However, Jack Parkinson marked his return from a fractured metatarsal by sweeping home Welling's second goal from Tony Finn's corner in the 88th minute.
Dean Cracknell finally confirmed the points deep into injury time when he converted a Welling counter attack.
Welling should have been well ahead by half-time but Sam Hurrell, Finn and Whitnall were all denied by the visiting. Weston's only chance was the best of the game but Josh Llewellyn blazed wide.
Cracknell hit the post from 25 yards within 30 seconds of the second half but Welling eventually made their dominance count to continue their climb up the table.
Former Bromley striker Ali Chaaban scored a last minute penalty for Staines as the Blue Square South match between the two sides ended 2-2 on Saturday.
Staines had taken the lead through Richard Butler after 13 minutes, before Bromley hit back with two goals.
Gareth A Williams equalised on the quarter of an hour mark before Rob Gillman struck after 32 minutes to put them in front.
Just when it looked as though the visitors would take all three points they were denied by Chaaban’s spot-kick.