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.

An 18-year-old woman was driving a Toyota Yaris when it was in a crash with a Ford Transit van on the A595 near Bootle at 19:55 BST on Friday.
She and her front seat passenger, also 18, were pronounced dead at the scene.
A third 18-year-old woman, in the back of the car, was flown by air ambulance to hospital where her condition is said to be "critical".
The 51-year-old man driving the van was also flown to hospital, where he is in a serious but stable condition.
The road was closed for six hours for the vehicles to be examined and then removed, Cumbria Police said.
The force is appealing for witnesses.

In previous years crime figures have seen a significant rise during the annual celebration.
Good weather on Saturday meant a larger number of spectators than usual turned out to the event.
However, Insp Stuart Wilson said that other than a "few minor incidents" it had passed peacefully.
He said: "The annual Guid Nychburris celebrations started early with police cyclists escorting about 200 horses through the traffic in Dumfries.
"The good weather meant that there were a large number of spectators out to watch the horses, the crowning ceremony and the parade in the centre of Dumfries but it was a good family atmosphere and there were no policing issues.
"Local police were assisted by police horses and Operational Support Division from Glasgow to make sure the crowds were safe."
Sean Raggett's second-half strike earned promotion hopefuls Dover a hard-fought win over relegation-threatened Welling in the National League.
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Two teenagers have been killed and another seriously injured in a car crash in Cumbria.
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Police have praised the "good family atmosphere" among the thousands of people who turned out to watch Guid Nychburris events in Dumfries.