This results means that Wasps move up to second place in the table and odds-on for a place in the end-of-season play-offs, while Bristol are now firmly rooted at the bottom of the standings.
But it was actually Bristol who started the stronger side in this encounter, Shane Drahm kicking a penalty in the fifth minute to give his side a 3-0 lead.
Indeed, Bristol could have stretched that lead by three points some moments later when they were awarded a second penalty but Drahm could not find his range.
The visitors were in confident mood, however, and seemed determined to achieve victory with ball in hand as they spurned kickable penalties to look for try-scoring opportunities.
They eventually found an opening in the Bristol defence through centre Stuart Abbott, the try converted by Alex King.
Bristol struck back through two further penalties from Drahm and the score remained at 9-7 at the break.
The second half saw Wasps hit their rhythm as their running game came together to devastaing effect.
Wasps hit Bristol with two quick-fire scores from flanker Paul Volley and England fullback Josh Lewsey, before Mark van Gisbergen romped over for his side's third try.
While Bristol were able to claw back three points through a penalty kick by Drahm, there was more punishment to come for the home side in the final quarter.
Centre Ayoola Erinle went over for Wasps' fourth try and back rower Joe Worsley sealed the victory with a late try.
The Shoguns did gain some consolation with a late try from winger Brendon Daniel, but the beleaguered West Country men now look on the fast track to oblivion.
For Bristol Shoguns:Try: DanielCon: DrahmPens: Drahm 4
For London Wasps:Tries: Abbott, Erinle, Lewsey, Van Gisbergen, Volley, WorsleyCons: King 2