The Dragons started the match brightly, slotting an early penalty and following that up with a try to open up a 10-0 lead after 13 minutes.
Bristol hit back in fast and furious fashion, crossing for three tries in quick succession and adding a penalty to complete a 10 minute 24 point blitz as they headed into the halftime break with a 24-10 advantage.
Bristol got off to the perfect start in the second half, with an early try extending their lead out to 31-10. The Dragons needed a response and did just that with their second try to get back into the match at 31-17 with half an hour to play.
Bristol kept the scoreboard ticking over, kicking two penalties to push their lead out to 20 points with 10 minutes remaining.
Bristol would seal the match in style, as they cut loose with some superb running rugby to race in three tries in as many minutes to march on to the Challenge Cup Semi Finals where they will face the winner of Bordeaux v Edinburgh.
SEMI AND BEN IMPRESS
Semi Radradra continued his incredible form for Bristol ripping through the Dragons' defence whenever he had the opportunity to do so.
The Fijian made the break for Ben Earl's first try of the game as he sped past a few defenders into a gap before sending the ball out wide. Earl ran an excellent line from the set piece to receive the final pass to score.
Earl was named man of the match as he made over 40 running metres and dotted down twice.
The 22-year-old made his debut during this year's Six Nations and had two keen on lookers with both Eddie Jones and Warren Gatland in the stands at Ashton Gate.
The Saracens' loanee ran excellent support lines throughout the game and also got stuck into the little defence Bristol were asked to do. He made seven tackles and won a turnover for the Bears in his 80 minutes.
Radradra directly assisted Earl's for his second try as the Fijian stepped off his right foot, outside the Dragons' 22, and drew in three defenders before offloading the ball out to Earl who showed off his footwork before going into score.
Radradra also showed off his excellent support lines for his try as he was on the end of a lovely offload from Callum Sheedy as he powered through a gap.
The centre looked as if he was searching for support before deciding he was better of having a go himself as he evaded the two Dragons' defenders to score an excellent try.
Dan Thomas also made telling impact off the bench to grab a brace with his and Bristol's final try proving to be arguably the best of the night.
Pat Lam's men now march into the semi-finals of the Challenge Cup for the first time where they will either met Edinburgh or Semi Radradra's former employers, Bordeaux.