The champions came to the Rec at near full strength for the first time since accepting their 35-point deduction and £5.4million fine for breaching salary cap rules.
Bath had plenty of territory but lacked the kind of quality possession that Saracens created for the game's only try, finished off by wing Sean Maitland on 33 minutes.
Maro Itoje and Billy Vunipola were also making their first appearances for Saracens since the final on November 2, while Elliot Daly made his debut at full-back following his move from Wasps in the summer.
The home side, hampered by knee injuries to skipper Charlie Ewels, England wing Anthony Watson and number eight Zach Mercer, handed the captaincy to World Cup winner Francois Louw. Sam Underhill, another of the tournament's eye-catching performers, joined him in the back row.
Disappointing in the first two rounds of the Heineken Champions Cup, they were looking to recover the form that yielded morale-boosting home wins over Exeter and Northampton. They had won the last three meetings in the league against Saracens.
Priestland kicked Bath into an early lead after Mako Vunipola was penalised at a Saracens scrum put-in. Farrell levelled the scores on 10 minutes only for Priestland to kick his side back in front from short range.
With referee Karl Dickens strictly policing encroachments on the offside line, Farrell made it 6-6 on 17 minutes.
Bath generated a lot of the steam rising off the two packs in the cold night air. They won plenty of rucks but never slickly enough and the only reward for a long spell of pressure was a third penalty for Priestland after Daly's no-arms tackle on Semesa Rokoduguni.
The 9-6 lead was short-lived as Saracens suddenly added power and pace to their game and Itoje made a telling burst in midfield. Quick ruck ball found Maitland in space on the left wing and the Scotland international touched down after a sublime sidestep.
Farrell converted and added a 40-metre penalty which gave the visitors a 16-9 advantage at the break.
There was a lucky escape for Bath soon after the restart as they tried to run the ball out from their own line and Rokoduguni's clearance kick was charged down. Saracens kept the pressure on and Farrell landed his fourth penalty.
Full-back Tom Homer offered a glimpse of attacking potential with a searing 60-metre break but Saracens scrambled back to kill the threat. Priestland and Farrell then swapped penalties again to make it 22-12.
A tiring Bath were being regularly penalised at the breakdown, whether in possession or defending, and Farrell landed a sixth penalty.
A yellow card for cynical play ruled Itoje out of the last nine minutes but Bath could not earn even a losing bonus point.