Owen Farrell landed a penalty via both posts with the last kick of the game to earn Saracens the points in a pulsating finish at the Recreation Ground.
The lead changed hands five times in the second half as Farrell's kicking regained the initiative for the Aviva Premiership Rugby champions only for Sam Vesty to edge Bath back in front with a penalty six minutes from time.
The drama of the last few minutes saw Neil de Kock have a clearance kick charged down on his line but Bath could not apply the killer blow and Saracens cleared from behind their own posts.
With only seconds left on the clock, the Londoners won turnover ball and Brad Barritt attacked down the left to win the final penalty from referee Greg Garner. Farrell finished with a tally of 23 points.
The hosts won the try count 2-1 thanks to Nick Abendanon and Carl Fearns while James Short crossed for the visitors' score but a losing bonus point will be of little consolation.
Bath took the field without skipper Stuart Hooper, ruled out with a bruised arm, so Dave Attwood made his Aviva Premiership Rugby debut in the second row after a summer move from Gloucester. David Flatman took over the captaincy as the home side looked to build on last weekend's encouraging 22-9 win at Newcastle.
Saracens' starting XV, led by former Bath captain Steve Borthwick, were unchanged from their 'home' defeat by Wasps at Twickenham.
Unusually for the Rec, the attendance was more than 2,000 short of capacity but Vesty wasted no time in raising cheer. The fly-half chipped over a second-minute penalty after Alex Goode was hunted down by home flanker Fearns and the Saracens full-back's team-mates were caught off their feet at the ruck.
The champions quickly fashioned their first try-scoring opportunity at the other end but Michael Claassens made a timely tackle on wing Short and Saracens had to be content with a Farrell penalty.
On 11 minutes the left wing made amends after a mix-up in the Bath defence allowed him to stroll over unopposed. Farrell added the conversion.
Almost immediately it was Short's turn to look on helplessly as Vesty's grubber kick through bounced kindly for Abendanon to run in from 20 metres. Vesty kicked the conversion to level the scores at 10-10 after 14 minutes.
Saracens' advantage in the scrum allowed Farrell to pull back three points and Bath's effort was being undermined by a string of penalties. Only the visitors' handling errors and outstanding tackling by Fearns kept them at bay. Vesty was wide with a straightforward kick on the half-hour but six minutes later Bath broke out to score a spectacular try through Fearns.
Dan Hipkiss and Claassens made progress up the left and Ben Skirving broke free to send his back-row colleague galloping 40 metres to the line. Paul Roberts, deputising for Vesty in the blood bin, landed the conversion to give Bath a 17-13 lead at the break.
Saracens came out for the second half in more determined mood, using Farrell's rolling kicks to establish position. Bath obliged by conceding penalties and the young fly-half was on target three times in seven minutes to regain the lead at 17-22.
Bath upped the intensity in response and Vesty pulled back three points on the hour mark. Suddenly it was Saracens' turn look vulnerable as the home side poured into the visitors' 22 and flanker Andy Saull took a yellow card to prevent a likely try. Vesty put Bath back in front at 23-22.
The see-saw scoring continued as Farrell kicked his side into a 23-25 lead with seven minutes remaining only for replacement prop Mako Vunipola to needlessly obstruct Tom Biggs at the restart. Vesty's nerve and his kick was good. A frantic finish saw Charlie Hodgson send a drop-goal attempt wide before Saracens' great escape from their own line to earn that last penalty chance.