Their top scorer in the league is Darren Pratley, with only seven so far.
Swansea's top-scorer Pratley eventually settled nerves in the 72nd minute with his seventh of the season.
Swansea equalised moments before half-time when Darren Pratley's chip picked out Cedric van der Gun to flick home.
Swansea could even have taken the lead moments later but Pratley was flagged off-side as he forced home Sinclair's cross-shot.
It was no less than the Swans deserved after taking the game to Cardiff with Pratley and Dyer looking particularly lively down the right.
But superb Pratley ensured Swans extended their unbeaten run to 10 with a fine volley just before the hour as the hosts held on.
Moments later, Pratley took an inswinging corner and Sinclair was given the time and space to strike the ball narrowly wide from 20 yards.
Darren Moore's header from a corner gave Barnsley hope after slack Swansea defending, but Pratley secured the win following a fine Andrea Orlandi set-up.
Nathan Dyer's header from Pratley's free kick broke the deadlock on seven minutes before Pratley himself added a second six minutes later with a volley.
Luke Steele parried a Darren Pratley header into Ashley Williams for the opener, then Shefki Kuqi glanced in a diving header off Mark Gower's cross.
Then Dyer linked up brilliantly with Pratley who put in the former Southampton winger to strike at goal, but Jason Steele dived low to deny him.
The second goal finally arrived via Pratley in the 72nd minute as the in-demand ex-Fulham trainee capped a driving run with an unstoppable shot past Jake Cole.
Lee Frecklington put a header wide for the Posh who rarely threatened and the Swans were more worried by injuries to midfield duo Bodde and Darren Pratley.
Rangel played in Darren Pratley who cut back from the by-line to Sinclair but his faint touch was not enough to steer the ball home from six yards.
Pratley had not scored for 26 games since last season's corresponding fixture at the Liberty Stadium but he certainly proved Swansea's hero on the hour to seal the win.
Swansea started strongly and had their first chance just three minutes in when Pratley's flick found Craig Beattie but the record signing could not beat David Marshall in the Cardiff goal.
Pratley cut inside and spotted Van der Gun coming in off his wing on the right flank, dinking a cross into the path of the onrushing Dutchman to head home from six yards out.
And within a minute, the visitors were behind when Bessone's left-hand cross at the other end found Pratley at the near post to volley home his second of the game and put the Swans back in front.
Swansea continued to threaten and Pratley was at the centre of the action and it was his ball that found Welsh midfielder Joe Allen on the edge of the box, but this time Marshall parried to safety.
And the Swans doubled their lead six minutes later as Bessone crossed from the left and Beattie appeared to be bundled over in the box but it mattered little as Pratley pounced on the loose ball to fire home.
The loss of captain Garry Monk in the warm-up to a calf injury turned out to be a blessing in disguise as Fede Bessone, his replacement, set up both Pratley's goals for Swansea's first derby league win in 10 years.
It did not matter as the home side were ahead seven minutes later when Pratley's free-kick from by the corner flag was flicked on by Ashley Williams and Dyer was unchallenged to head into an empty net with Marshall nowhere.
Swansea's stunning win was also in front of a record Liberty Stadium crowd of 19, 309 that were treated to a footballing lesson of how to dominate your opponent - and Swansea had star midfielders Leon Britton and Darren Pratley on the bench.
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