A Nils Mordt penalty then got Saracens on the board before he converted Kameli Ratuvou's try as the hosts gave themselves a 10-5 advantage.
Sale were awarded a penalty try midway through the half which allowed Davies to add the extras and make the score 12-10.
Another Mordt penalty put Sarries in front by a single point and their lead was increased thanks to Joel Tomkins's try minutes before the half-time whistle.
Sale cut the deficit to a single point after the break when Fraser McKenzie's try was converted by Davies and a Danny Cipriani penalty made the score 22-20 to the home side with nearly an hour played.
Ben Spencer's kick temporarily put Saracens in front, only for Cipriani to score his second penalty with seven minutes left to play to give Sale the victory.