Sean Marsden: Yellow cards turned the game
Bristol coach Sean Marsden accepted that it had been a tough time for the club in a week which had seen the suspension of their director of rugby Andy Robinson.
Bristol suspended Robinson following their dismal start to the season and appointed Mark Tainton as acting head coach.
Despite the 31-25 defeat, Bristol produced a committed performance and were unlucky not to pick up their first win of the season after nine consecutive defeats.
Bristol had deservedly built up a 13-5 lead but the sin-binning of full-back Jack Wallace and hooker Rob Hawkins early in the second half proved crucial as Worcester scored 28 unanswered points before the visitors rallied.
Giorgi Nemsadze, Tom Varndell and Wallace scored their tries with Billy Searle converting two and adding two penalties.
Marsden said: "It was a difficult week and all we could do is to focus on tonight's game and I thought we showed character and belief. We are ultimately disappointed that we didn't win the game.
"We played with a lot of pace in the opening 40 minutes and should have been further ahead at half-time.
"The two yellow cards were the turning point but we defended our line well and came back strongly so credit to the lads.
"We are going to grow as a group and I firmly believe we are going to kick on and hopefully win next week."
Worcester capitalised on Bristol's errors to score five tries for a bonus-point win with Max Stelling and Josh Adams both scoring two. Debutant scrum-half Michael Dowsett scored the other with Jamie Shillcock adding three conversions.
Warriors coach Simon Cross said: "It was frustrating that we put ourselves under so much pressure late on in the game but I'm delighted we found a way to win.
"It was a scrappy game and I'm thrilled at the way we scrapped as our set-piece was under pressure in the first half, which always seems to happen against Bristol.
"However we are disappointed with our accuracy in their 22 especially when they were down to 13 men."