© PA Photos
A quiet first half saw the sides go in 6-3 at the break, Gavin Henson kicking two penalties for the home side.
Stephen Donald levelled for the visitors in the 51st minute, and put them ahead 10 minutes later.
It looked like finishing 9-6 to Bath before Welsh were awarded a penalty with less than two minutes remaining.
Gordon Ross kicked the penalty to level the scores again, before Scott scored the game's only try in the last minute of the game.
The result doesn't affect either side's league position, but does see Welsh close the gap to Worcester Warriors.