Banaham's late try proved in vain as Ulster's arduous journey ended with a win
Ulster took a step closer to qualifying for the Heineken Cup knockout stages after all but eliminating Bath following a tense win at The Rec.
Bath took a 14-13 half-time lead after an early Matt Carraro try, along with three penalties from Olly Barkley.
But a second-half Nevin Spence score added to Adam D'Arcy's try and the boot of Ian Humphreys put Ulster in front.
Bath responded with a superb length-of-the-field Matt Banahan try but the hosts could not find the extra points.
The defeat was Bath's third in four matches in this season's competition and means Steve Meehan's team face an almost impossible task to progress into the knockout stages of the tournament.
Ulster's Dublin detour to Bath
In contrast, Ulster boosted their prospects of finishing top of Pool Four with their third win of the campaign, with a home match against leaders Biarritz coming up next month.
The Magners League team, who trail Biarritz by three points, can top the group with a victory against their French rivals, who visit Ravenhill in January.
Volunteers shovelled snow off the stands and gangways to ensure the Recreation Ground was fit to host the match, which had been in danger of being called off because of the adverse weather conditions.
Ulster had been forced to travel from Dublin after flights from Belfast were suspended because of the snowy weather, although the pitch had been protected all week by a huge canopy-like covering.
Bath were 5-0 in front in the first minute when Lee Mears sent winger Carraro through into the corner, but D'Arcy responded when Ulster ran a 10th-minute penalty.
Humphreys added eight points with the boot, while Barkley kicked three penalties to leave the game evenly poised at the half-time interval.
But the momentum of the match changed when fly-half Butch James, making his first appearance in a Bath shirt since shoulder surgery in August, was yellow carded at the start of the second half for a scuffle with Dan Touhy.
The Springbok's absence allowed Spence to touch down for a 56th-minute converted try before Humphreys landed a 50m penalty to give the visitors a 23-14 lead.
However, Bath hit back in spectacular style 11 minutes from time with a try which started from deep inside their own 22m line.
Nick Abendanon's powerful burst cut through Ulster's defence before the ball was farmed out to the wing via a string of outlandish long passes, allowing Banahan to touch down in the corner.
Barkley could not add the extras from the touchline and an offence at the breakdown allowed Humphreys the opportunity to wrap up the victory with three points in the final five minutes.
Bath coach Steve Meehan: "It's very frustrating. The Heineken Cup is a competition in which if you don't play for the full 80 minutes, you're not going to win your games.
"We gave them the opportunity to score 10 points in the first half when we should have cleared our lines. "We are well aware that if we had played throughout with the intensity we showed at times, we should have run out comfortable winners. For whatever reason, that's just not happening."
Ulster head coach Brian McLaughlin: "In the first few minutes we looked a bit ropey but we clawed our way back to 14-13.
"Actually, I didn't think we played that well. Bath put us under ferocious pressure at times. We upped the intensity, though, and we cut our cloth a bit in the second half, playing a wee bit more the percentages. "We knuckled down and knew what we had to do. If we had control we had a chance.
"I said to the players afterwards: 'We've done nothing yet'. We've just got to keep winning."
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