The 22-year-old headed home from a corner as he scored just his fourth goal in 33 appearances this season.
Former Whites striker Michael Bakare had handed the away side the perfect start, curling the ball into the top corner in the second minute.
Stefan Payne scored at the near post to level before Raggett's winner.
Dover are three points behind Tranmere in sixth, with three games in hand, while Welling remain nine points adrift of safety.
Sean Raggett's second-half strike earned promotion hopefuls Dover a hard-fought win over relegation-threatened Welling in the National League.