Sinclair notched his sixth goal in the last four games to put his side on the way to their first away win of the season, ending Watford's four-game unbeaten streak in the process.
The former Chelsea man superbly fired home from Joe Allen's pass, with Stephen Dobbie doubling their advantage on the half-hour mark and Franck Nouble seemingly putting the game to bed in the second half.
It was a disappointing night for Watford, who had won their last three games, but late goals from Troy Deeney and Martin Taylor gave them hope of salvaging an unlikely point.
They thought they had done so in injury time but the offside flag denied John Eustace and spared the Swans, who move level on points with the Hornets as both teams remain in the early play-off picture.
Brendan Rodgers' men travelled to Vicarage Road without an away win in the Championship since they last visited the Hornets in March last season, but their impressive start belied their woe on the road.
And they went ahead with their first meaningful attack of the match when in-form Sinclair latched onto Joe Allen's long pass to fire home in emphatic style in the 11th minute.
Swansea continued to dominate proceedings and doubled their lead on the half-hour, with Allen again the creator.
He found space on the left and produced an inviting cross for Dobbie to fire high into the net from 12 yards.
If Watford were to get anything out of the game they had to try and mount a quick recovery and get a goal back before half-time.
They could have pulled one back within a minute but Adrian Mariappa's header was cleared off the line by Alan Tate.
Marvin Sordell and Danny Graham both went close as the Hornets desperately sought a way back into the game, but the Swans got to the half-time whistle unscathed and could have gone even further ahead, but Sinclair had a shot blocked.
Sinclair was a constant thorn in Watford's side and went close from 20 yards but his shot went just the wrong side of the bar.
Watford were unable to build any sort of momentum after the break and the Swans made sure the three points were going home with them when substitute Nouble got his name on the scoresheet with his first touch of the match.
Nathan Dyer broke free down the right and crossed for Nouble, who had the easiest of tap-ins from six yards and celebrated his first senior goal.
Watford then enjoyed their best spell of the game and Graham should have reduced their arrears but shot wide from point-blank range, before Malky Mackay's men did eventually register when Deeney nodded home from close range with 14 minutes remaining.
Hopes of an unlikely comeback were raised further with five minutes remaining when Taylor headed home Lloyd Doyley's cross.
And the Hornets thought they had completed their task deep into injury time, but Eustace's goal was disallowed for offside.