Despite the triumph, Dai Young's side have yet to discover whether they have earned themselves a home quarter-final in the competition.
© David Davies / PA Photos
Daniel Evans then touched down for Newport, with Prydie adding the extras to give his side a 10-3 advantage.
A second Robinson penalty cut the deficit and Wasps went ahead for the first time in the match in the 36th minute when the 31-year-old added the extras to a penalty try.
However, the Dragons had the final say of the first half courtesy of another Prydie kick.
Wasps stretched their lead to 20-16 just before the hour when Robinson converted his own try.
Steffan Jones came off the bench to kick the home side to within a point of their opponents with five minutes left to play, but they were unable to find another score to snatch the victory.