But Worcester dominated the second period, with hooker Nick Seymour and the giant Semisi Taulava adding tries.
Tiff Eden converted the first two tries with replacement Elliott Davies knocking over the third conversion and adding a tricky late penalty to give the visitors victory.
Academy Manager Mark Hewitt said: "Josh ran in a fine try but Bristol were the better side in the first-half. We were playing against a strong breeze and they dominated in the scrum.
"But we came out fighting in the second-half. We were really strong, controlled the ball in the right areas and took our chances before Elliott sunk a fantastic penalty late on."
Captain Cam Goodhue said: "It was a tough match in the mud but the boys played really well, especially Nick Seymour who put in a fine shift."
Cavaliers will next be in action against The Army at Sixways on Wednesday 12 March.