Ireland fly-half Jonathan Sexton kicked six penalties - three during a 15-minute spell in the second period - while his two clinching goals arrived during the closing seven minutes.
Bath went close to collecting one of their most notable Heineken Cup scalps in recent seasons when England wing Matt Banahan barged over from close range after 65 minutes.
Centre Olly Barkley, who slotted two first-half penalties for a 6-0 lead, added the conversion to edge Bath one point in front, but Sexton nudged Leinster towards a priceless Pool Three away win when home skipper Francois Louw infringed.
And to complete a miserable two-minute spell for South African Louw, he was then sin-binned as Bath ended the game a man short.
Bath now head to Dublin for next Saturday's return fixture and, while they are not mathematically out of quarter-final contention, it will require a Herculean effort to turn things around, with a losing bonus point providing scant consolation.
Their fellow English challengers Northampton already look set to bow out before the knockout phase, and with Bath, Gloucester and London Irish also scrapping - it would appear in vain - to progress, this season's Heineken Cup could feature very few realistic Aviva Premiership contenders.
Bath's injury problems showed no sign of abating, with their list of absentees including flanker Lewis Moody, hooker Lee Mears, lock Stuart Hooper and wing Tom Biggs. New signing American World Cup front-row forward Chris Biller made his full debut.
Leinster, though, paraded a familiar galaxy of stars - long-term injury victim Brian O'Driscoll excepted - as they targeted a victory that would put them in pole position for quarter-final status.
The game started at a frantic pace with Bath enjoying the lion's share of possession, but Leinster soon displayed their attacking prowess when Luke Fitzgerald counter-attacked from inside his own 22 before the move was halted 60 metres downfield.
Visiting full-back Rob Kearney then sparked another daring raid, only for flanker Sean O'Brien to knock on in midfield, yet Bath's forwards did not shirk from their task and ensured sustained territorial dominance.
Barkley booted Bath into a 16th-minute lead, and with the home back-row of Louw, Guy Mercer and Simon Taylor offering repeated ball-carrying threats, Leinster had their work cut out defensively.
But Leinster's organisation in that department kept Bath out, restricting them to a second Barkley penalty six minutes before half-time.
Leinster, for all Bath's possession, still looked the more dangerous team with ball in hand, and O'Brien almost broke through as the interval beckoned after Sexton had missed a touchline penalty attempt.
Bath snuffed out the danger courtesy of wing Matt Banahan's tackle, but they infringed again deep into stoppage time and Sexton made no mistake from just 25 metres out, halving Leinster's deficit as they trooped off 6-3 adrift.
Ireland prop Cian Healy was introduced for the second period, replacing Heinke van der Merwe, and Leinster immediately established momentum after a surging run by an ever-dangerous O'Brien.
O'Brien was then guilty of wasting a golden chance for Leinster, ignoring a three man overlap after wing Isa Nacewa's pace and awareness had set up the opportunity, and Bath escaped.
Pressure, though, was mounting on the Bath defence after Sexton's second penalty hauled Leinster level, and then he completed his hat-trick to make it 9-6 after 54 minutes.
Another Sexton penalty, kicked after Leinster prop Mike Ross was held up over Bath's line, looked to have given Leinster a commanding lead in an attritional contest.
Bath, though, had other ideas, and they hit back 15 minutes from time when slick work by their backs ended in Cuthbert delivering a try-scoring pass to the powerful Banahan.
Barkley landed the conversion to edge Bath in front and set up a rousing finale between two evenly-matched teams that had gripped a capacity Recreation Ground crowd of 12,200.
But Louw infringed and Sexton did not require a second invitation to seal the deal as Leinster stayed on course to possibly reach their third European final in four seasons.
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