New Bath players, Samoan No.8 Isaac Feaunati and former Bristol wing Brendon Daniel, both scored tries, while fullback Matt Perry and centre Alex Crockett scored five pointers, giving the visitors five points in their first game of the new season, after narrowly avoiding relegation in 2002/03.
Inside centre Olly Barkley contributed 18 points with his boot, while fly-half Chris Malone, recalled from a loan spell at Bristol, slotted one drop-goal.
Bath scored their fourth, and bonus-point clinching try in the 57th minute through Crockett, but Leeds do deserve credit for managing three of their own in the remaining minutes, replacement halfback Alan Dickens, No.8 Colm Rigney and discarded England wing Dan Scarbrough all crossing and earning the home side a bonus point.
The visitors ran into a useful 17-0 lead after just 25 minutes of play, Barkley's drop-goal and a solitary penalty from Tykes No.10 Duncan Hodge the only other scoring in the first stanza – and leaving Bath ahead by 20-3 at the break.
Barkley slotted a penalty early in the second half to extend that lead to 23-3 before Daniel dotted down his first try for Bath, Crockett following in the 57th minute, but not before David Rees opened Leeds' tally with a fine effort in the 51st minute.
Hodge converted and kicked a penalty, but Crockett's try – and the successful conversion – left Bath ahead by a massive 32-13.
But Leeds finally pulled it together and their three tries proved that they can, indeed, play, only for three Barkley penalties in the final 11 minutes to keep Bath ahead by nine points – robbing Leeds of a second bonus point.
For Leeds Tykes:Tries: Rees, Dickens, Rigney, ScarbroughCons: Hodge 3Pens: Hodge 2
For Bath:Tries: Feaunati, Perry, Daniel, CrockettCons: Barkley 3Pens: Barkley 5DG: Malone