Corsham's Tom Hankins
The Swords had triumphed by five wickets in the reverse fixture the previous week but found themselves on the ropes after they collapsed from 165-5 to 183-9 chasing 218 to win.
Bath looked on course for victory as top-scorer Lloyd Davies (63) and Charlie Duffell (44) compiled a century stand for the fifth wicket. But they were separated when Wiltshire seamer Alex Senneck bowled Davies on his way to superb figures of 5-30 from 16 overs and Duffell fell to Rich Foley four runs later.
When Senneck bagged a catch off Nick Harrison (2-48) to dismiss number nine Adam Kelly (6), Bath looked in real danger but Marchant (8no) and captain Thorburn (1no) dug deep and Bath's 55 overs ended the score on 197-9, giving Corsham a winning draw.
Davies and Duffell had helped Bath recover from a poor start which saw them reduced to 57-4.
Earlier, Corsham's innings had begun in even worse fashion after Thorburn asked them to bat first, openers Malinda Thotiwilage and Peter Dennett both departing without scoring, leaving the visitors 2-2.
The demise of ex-Bath batsman Tom Hankins for 12 brought Foley to the crease and he proved a key figure, putting on 51 for the fifth wicket with Dan Collier (25) and then sharing an unbroken 110-run stand with Senneck, who hammered 62 off 80 balls including three sixes. Foley finished on 59 not out as Corsham posted 217-5.
The result further dented Bath's chances of marking their 150th anniversary by retaining the championship and they lie off the pace set by Taunton St Andrews – who visit Corsham on Saturday – in fifth spot.
Thorburn's men face the league leaders away in the last 16 of the ECB National Club Championship on Sunday after hosting lowly Knowle 24 hours earlier.
Opener James Golding hit a superb 112 to help Keynsham chase down hosts Cheltenham's hefty total for an eight-wicket victory in Premier 2.
Keynsham had been beaten by 87 runs by the same opponents on the previous Saturday and were handed a tough task when Cheltenham racked up 287-6 off their 50 overs.
Jackson Thompson hit 58 off 42 balls while Kyle Stovold top-scored with 66 and having put on 110 for the sixth wicket alongside John Griffiths (44no). Former captain Simon Knowles was Keynsham's most successful bowler, taking 3-58.
Graham Williams (24) and Golding gave Keynsham a solid start in reply by putting on 76 for the first wicket.
Skipper Richard Pannell then joined the former Kent all-rounder and the pair shared a match-defining stand of 145 before the former exited for 82 off 86 balls. Dave Burston also scored quickly for his 30 off 34 as he and Golding finished the job with four balls to spare.
The victory saw Keynsham, who are fourth, close the gap on third-placed Cheltenham to four points. They entertain second-from-bottom Bristol West Indians on Saturday.