A late try from Kalaveti Ravouvou levelled the scores at Ashton Gate, with James Williams adding the conversion to seal the points for Bristol.
The Bears moved up to 33 points, level with Northampton, who host Harlequins on Saturday. It was a sixth successive away defeat for Salerugby, who remain seventh with 19 points.
Wing Tom O'Flaherty dotted down first for Sale, but lock Joe Owen responded for the home side to make it 7-7 at the break.
Ernst van Rhyn touched down for Sale early in the second half, and George Ford added a penalty to take Sale 17-7 up.
Back came Bristol, though, with Matias Moroni going over before Ravouvou charged down the left and cut back in field to touch down under the posts.
Newcastle held off a spirited comeback from fellow strugglers Gloucester to win 25-19 at Kingston Park - their first victory in nine outings this season.
George McGuigan opened the scoring for Newcastle and, with both teams reduced to 14 after Jamie Hodgson and Arthur Clark were sin-binned, Tom Christie bagged the hosts' second try before Ollie Thorley replied for Gloucester.
Brett Connon's penalty gave Newcastle a 15-5 lead at the break, and following Simon Benitez Cruz's try after the restart, Gloucester's evening soured when Thorley was shown a 20-minute red card, and Connon kicked another penalty.
Gloucester threatened a late comeback with tries from Mikey Austin and Ciaran Knight, but Newcastle held on for a vital win.
Saturday's other big game sees Bath hosting Exeter, while Leicester host Saracens on Sunday.