Tom Arscott's 12th-minute try gave Bristol the lead, and although Loki Crichton landed a penalty for Worcester the visitors led 5-3 at the break.
The hosts went in front on 53 minutes through Pat Sanderson's converted try.
And they wrapped up an important victory as ex-All Blacks lock Greg Rawlinson and Dale Rasmussen both went over in the final quarter.
Hill on England hooker Mark Regan:
"I have not asked the physios what his problem is yet but it looked like a neck or head. He's moping about in the dressing room at the moment. "He would not have come off if he was not 100% so it depends on how he reacts now (on his condition with England). It's not what he would have wanted next week."
Worcester: Delport, Gear, Rasmussen, Tuitupou, Benjamin, Crichton, M Powell, Morris, Lutui, Tuamoepeau, Rawlinson, Gillies, Hickey, Sanderson, Horsman.
Replacements: Black, Mullan, Bowley, Talei, R Powell, Brown, Pennell.
Bristol: L Arscott, T Arscott, Higgitt, Hill, Elliott, Strange, Beveridge, Clarke, Regan, Crompton, Winters, Hohneck, Salter, To'oala, Blowers.
Replacements: Linklater, Hobson, Llewellyn, El Abd, O'Riordan, Taumalolo, Robinson.
Referee: Tony Spreadbury (Bristol)
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