A quiet opening spell saw the visitors edge ahead through two Nicky Robinson penalties.
However, a strong showing from Worcester allowed Chris Pennell and David Lemi to cross the try line, giving the hosts a 15-6 lead at the break.
Wasps made a bright start to the second half, with T Rhys Thomas scoring inside the opening two minutes, before Ashley Johnson added a second 10 minutes later to give the visitors the lead, with Josh Drauniniu in the sin bin for Warriors.
Blair Cowan and Jonny Arr added further tries for Worcester, giving the home side an 11-point lead heading into the final 10 minutes.
Wasps pushed hard for another try, Jack Wallace eventually crossing the whitewash after the 80 minutes had expired.
The win leaves Worcester ninth in the table, while Wasps sit fifth.