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Joe Healy's stoppage-time penalty preserved Welling's impressive home record in Blue Square Bet South on Tuesday.
The Wings looked to be heading for only their second league defeat at Park View Road in 19 months until Healy came to their rescue.
It was no more than they deserved for a rousing second-half fightback after Dover dominated the opening 45 minutes.
Referee Ian Cooper, who was a key figure throughout, booked the first of seven Whites player as early as the 10th minute when Tom Wynter tripped Theo Fairweather-Johnson.
Two penalty appeals had already been ignored by Cooper but it was third time lucky for the visitors when Ben Martin was penalised on the half-hour. The impressive Ben May confidently tucked away the spot-kick to put Dover ahead.
May had a header disallowed for offside early in the second half but it was Dover keeper Mitch Walker who threatened to emerge as their hero.
Having denied Healy, he then thwarted Lee Clarke's low drive with his legs.
Not to be outdone, Wings keeper Sam Mott produced a stunning reflex stop to prevent Martin scoring an own goal following May's left-wing cross.
Welling introduced former Whites winger Harry Baker and Jon Main late on and it was the latter that nearly went close when his close-range shot was diverted over the bar.
But, from the resulting corner, referee Cooper spotted a push by Walker and pointed to the spot - the keeper becoming the seventh yellow card of the night for the visitors.
Healy converted the penalty while Main nearly produced a grandstand finish for the hosts but curled his shot wide from the edge of the area deep into injury time.