John Cooney scored 12 points for the Pro14 side, touching down in the first half, as well as adding two conversions and a vital penalty which won the game.

Rob Lyttle also crossed for the visitors, while Gabriel Hamer-Webb hit back with a try on his European debut.

Freddie Burns, up against his brother Billy, kicked eight points with his boot, but it was not enough for Bath.

Bath looked set to clinch victory right at the death as they had an overlap but Ulster wing Jacob Stockdale produced a sensational interception to prevent Zach Mercer finding Semesa Rokoduguni with the pass.

On a heavy pitch at the Recreation Ground which left a number of players suffering with cramp, 1999 champions Ulster triumphed against the 1998 winners of this competition.

Cooney reacted quickest to charge down from Burns' kick under pressure, allowing a breakaway try for the visitors, as Burns' long-range penalty from close to the halfway line meant his side trailed 7-6 at the interval.

Ulster looked set to lead by four at the interval as Cooney had a penalty chance from almost right in front of the posts but incredibly, the scrum-half missed.

Lyttle's try from from Will Addison's kick further stretched the away side's lead, but 19-year-old Hamer-Webb side-stepped and darted in with his first touch of the ball just a few seconds after coming on to the pitch.

That effort came from Jonathan Joseph's pass, the centre one of three England players - including Sam Underhill and Ruaridh McConnochie - returning to the side after their World Cup exploits.

Welshman Rhys Priestland popped over a penalty to put Bath in front for the first time, but Cooney kicked his own with 10 minutes remaining before the dramatic finale as Ireland wing Stockdale produced his vital intervention.

Ulster host Clemont Auvergne in their next game on Friday (kick-off 19:45 GMT), while Bath travel to Harlequins on Saturday, 23 November (17:30 GMT).

Bath: Burns; Rokoduguni, Joseph, Roberts, McConnochie; Priestland, Chudley; Obano, Dunn, Stuart; McNally, Ewels (capt); Williams, Underhill, Mercer.

Replacements: Walker, Boyce, Judge, Stooke, Bayliss, Cook, Wright, Hamer-Webb

Ulster: Addison; Lyttle, Marshall, McCloskey, Stockdale; Burns, Cooney; McGrath, Herring, Moore; Henderson (capt), Carter; Matthew Rea, Murphy, Coetzee.

Replacements: McBurney, O'Sullivan, O'Toole, O'Connor, Reidy, Shanahan, Curtis, Ludik