Ulster won the battle of the former Heineken Cup champions at Ravenhill and gave Bath a huge mountain to climb if they are going to reach the knock-out stages after a narrow 22-18 success.
Fly-half Ian Humphreys landed five penalties for Ulster but it was their heroic pack, some excellent work by the backs and Bath's horrendous error count that put them through.
Bath had a touch of revenge to take care of, as they went down to the Ulstermen 26-12 at Ravenhill last year which signalled a miserable Heineken Cup campaign. And the visitors had to draft in England international centre Olly Barkley from the bench to make his 200th appearance for them as initial selection Shontayne Hape became a father.
Ulster, despite finishing second behind Stade Francais last season, also failed to qualify for the knock-out stages and, in almost a winner-take-all situation at Ravenhill, they were fired up to get as many points as possible to have a chance of going through this time.
Bath soon quietened the home crowd. Barkley put over the first penalty of the match and Ulster were stung by an interception try just as they looked to score themselves. Centre Paddy Wallace fed Ireland flanker Steven Ferris only for the back row to be brought down close.
Ulster hammered Bath's line but, when Humphreys threw a pass out on the wide left, visiting full-back Jack Cuthbert intercepted. Despite the attentions of former Bath wing Simon Danielli, Cuthbert just got home to touch down near the posts, making Barkley's conversion easy.
Humphreys booted a penalty as Ulster, through some stern work by their pack, fought back, causing Bath to make countless mistakes. That work was undone when a high ball was fluffed by home full-back Adam D'Arcy for England captain Lewis Moody to grasp and run in from 30 metres.
By half-time, though, Ulster were right back in it as Bath's penalty count went into double figures and loose-head prop David Flatman was sin-binned for a high tackle.
The second interception try of the game saw Ulster number eight Pedrie Wannenburg grab a loose pass by Simon Taylor on halfway and race over to score. Humphreys converted, and with a second successful penalty the Irishmen went into the break just two points down at 15-13.
Barkley restored the five-point lead but Ulster battered Bath and hit back with two more penalties to take a 19-18 lead. It was just what Ulster deserved for their effort when Bath infringed in front of their posts for Humphreys to boot his fifth penalty.
Bath battled for another try but found the home defence brickwall solid. And a late knock-on right on the Ulster line by Bath's pack sealed their fate and a left them a huge task to get back into the cup hunt this season.