Glenn Jackson's early penalty put the home side in front but they did not add to that until the 50th minute, by which time Bristol were well ahead.
Three penalties by Strange and a converted Marco Stanojevic try had put Bristol out of reach by half-time.
Jackson converted Ben Russell's try but Saracens had left it too late to win.
Bristol head coach Richard Hill: "They threw all their young lads on and they showed a lot of heart. I thought they were going to win the game but our defence was outstanding.
"Somehow we managed to hang on and that was an extremely important win.
"It takes a lot of doing when you are near the foor of the table to win away from home. That's a huge game for us."
Saracens: Bartholomeusz, Scarbrough, Sorrell, Harris, Vaikona, Jackson, Bracken, Lloyd, Byrne, Visagie, Raiwalui, Chesney, Randell, Seymour, Vyvyan. Replacements: Kyriacou, Broster, B. Russell, Armitage, Rauluni, Johnston, Powell.
Bristol: Going, Robinson, Higgitt, Cox, Stanojevic, Strange, Perry, Hilton, Regan, Crompton, Budgett, Llewellyn, Salter, Short, Ward-Smith. Replacements: N. Clarke, Nelson, Sambucetti, Lewis, Nicholls, Hayes, Denney.