Scott picked off Stephen Donald's telegraphed pass and had sufficient pace to hold off the cover for the match-winning try, which Gordon Ross converted.
It was the only try in a desperately poor game and a draw would have been a fair result as neither side played well enough to deserve a win.
Bath had the better scoring opportunities but though they went close on a number of occasions, they failed to take them. They had the more inventive backs but their scrum was continually under pressure.
Welsh were dogged throughout but provide no attacking spark and until Scott's effort, never threatened the try line.
Bath had an early chance for points but Donald was off target with a 40-metre penalty attempt.
Moments later, Welsh had their opportunity which they also failed to take as Gavin Henson was narrowly wide with his 45-metre kick - but with 10 minutes gone, the outside half was successful from 40 metres to put his side ahead.
Despite this reverse, Bath had the better of the first 15 minutes, desperate defence keeping the visitors out as Ben Williams and Michael Claassens made holes in the Welsh defence.
Against the run of play, the hosts broke out to win a penalty for Henson to extend their lead with a simple 25-metre kick.
The second quarter was more evenly contested, with the Welsh pack matching their opponents, but following a sustained drive from the Bath forwards the hosts were penalised and Donald made no mistake from 30 metres out.
The remainder of the half saw an abundance of penalty awards and a procession of aimless kicking, with Donald and Tom Arscott the worst culprits.
With the last movement of the half, Bath lock Dave Attwood did get over the line but he was held up by some stout home defence, allowing Welsh to retain their fortunate 6-3 lead at the interval.
Bath began strongly after the restart. Powerful runs from Williams and Matt Banahan gained them some momentum but the visitors lost a crucial scrum in front of the Welsh posts when they were easily pushed off their own ball.
However they did gain some reward when after 51 minutes Donald kicked his second penalty to tie up the scores.
With 20 minutes to go both teams made changes, with Henson and Banahan being the major casualties as they were replaced by Ross and Horacio Agulla respectively.
Exiles lock Kirill Koulemine was penalised for pulling down at the line-out and a well struck penalty from Donald put Bath into the lead for the first time.
With 10 minutes to go, Bath conceded a scrummage penalty. Ross' 45-metre kick was badly struck and fell short but with 90 seconds remaining on the, clock the outside half was successful from 30 metres to bring his side level.
Scott's heroics at the death then sealed the win.
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