It was the home side who took the lead when centre Sam Cox powered over after just four minutes.
Sarries wing Ben Johnstone ran in from 30 metres soon before fly-half Glen Jackson kicked over on 36 minutes to make the score 8-10 at the break.
The sides traded penalties before Ben Skirving touched down for Saracens while Jackson added a late drop goal.
Bristol head coach Richard Hill:
"I feel that we are marginally ahead of our target for this point in the season partially by virtue of Leeds going through a poor spell. "We needed Saracens to be a little off their game and for us to be at our
best to win these early matches when clubs are at full strength.
"When it comes to the pre-Christmas internationals and then the Six Nations
sides will be without some of their key players and that should help us.
"Saracens were very good at their set pieces and made very few errors on the
ball and it was always going to be tough to be sides like that."
Bristol: Stortoni; Robinson, Higgitt, Cox, Going; Strange, Perry; Hilton, Nelson, Crompton; Winters, Llewellyn; Salter (capt), El Abd, Lewis.
Replacements: Contepomi for Going (40), Rauluni for Perry (40), N Clarke for Nelson (70), Ward-Smith for Lewis (70). Not Used: Irish, Sambucetti, Gray.
Saracens: Castaignede; Johnston, Sorrell, Harris, Bailey; Jackson, Bracken; Yates, Cairns, Visagie; Raiwalui, Fullarton; Vyvyan (capt), Sanderson, Skirving.
Replacements: B Russell for Harris (69), Dickens for Bracken (61), Lloyd for Yates (60), Byrne for Cairns (60), Chesney for Raiwalui (55), Randell for Sanderson (55), Seymour for Skirving (82).
Att: 7,068
Ref: Rob Debney (Leicestershire).