The Swans were unchanged against a managerless Hornets side that had slipped into the relegation zone after Saturday's matches.
The corners came thick and fast for Swansea in the opening minutes but they were unable to make them count with Darren Pratley and Mark Gower both shooting wide when well placed.
In the tenth minute a flowing move saw Scotland shoot for the corner of the net but Watford goalkeeper Richard Lee dived low to push the ball behind.
The visitors were saved in the 22nd minute when a sliding tackle from Jay DeMerit dispossessed Scotland just as the striker was lining up a close-range strike.
Against the run of play the Hornets took the lead in the 33rd minute. Federico Bessone failed to clear and Lee Williamson played a neat one-two with Will Hoskins before slipping the ball into the net.
However, the Swans equalised straight from the kick-off when a defensive clearance fell to Bodde who thundered a 30-yard shot into the net with Lee floundering in the visitors goal.
Ten minutes into the second half there were loud appeals for a penalty as Watford's Tommy Smith appeared to handle in the box but the referee waved play on.
The Swans took the lead in the 59th minute thanks to Scotland. Bodde sent a superb 40-yard pass over to the striker who evaded the challenge of DeMerit and clipped the ball over the advancing Lee for a superbly taken goal.
The home side had time to complete a miserable afternoon for the visitors when substitute Bauza combined with Febian Brandy before slotting the ball into the right-hand corner of the net with the last kick of the match.