First-half goals from Ashley Williams and Shefki Kuqi set Paulo Sousa's side on their way and although Darren Moore pulled one back for Barnsley before half-time, Darren Pratley restored Swansea's two-goal advantage midway through the second period.
The Swans' grip on a top-six finish has not looked so assured in recent weeks with their latest defeat at Bristol City leaving them with only one win in their last eight games and just two points ahead of seventh-placed Blackpool.
Manager Sousa responded by making six changes to the side that lost last week. Argentinian full-back Federico Bessone returned after a two-month injury lay-off and Albert Serran, Kuqi, Cedric van der Gun, Leon Britton and Mark Gower all returned.
Martin Devaney, Moore, Iain Hume and Andy Gray were all recalled by Barnsley, whose season is petering out as they have now failed to win any of their last eight matches.
The early exchanges were highlighted by one of the misses of the season from Swans striker Kuqi, who pulled his shot from six yards wide in front of an open goal.
But the Welshmen were quickly into their stride, with Pratley forcing a smart save from Barnsley goalkeeper Luke Steele before Williams stabbed home his fifth goal of the season in the 23rd minute after David Cotterill's corner had been flicked into the six-yard box.
Steele then saved Gower's header at full stretch from a free-kick by Cotterill, who forced the visiting goalkeeper to pull off an even better save with a goalbound shot from 20 yards.
But Swansea maintained their momentum and Kuqi atoned for his howler in the 28th minute when he headed home Gower's cross from the right, low into Steele's left-hand corner to double the home side's advantage.
The Swans' two-goal cushion did not last long though when the visitors were gifted a goal courtesy of an error by Dorus de Vries.
The Swansea goalkeeper committed himself to Hume's corner, but failed to gather the ball and left the unmarked Moore with a free header at the far post.
Hume fired an early second-half warning for the visitors straight after the restart, flashing an angled shot just wide from the edge of the penalty area.
Barnsley manager Mark Robins then replaced Hume with 18-year-old striker Jamil Adam, who was making only his second senior appearance for the Tykes.
The visitors gave as good as they got in the second period. Van der Gun's shot from the edge of the area was well saved by Steele, but Hugo Colace went close for Barnsley with a 25-yard rocket which flew just over the crossbar.
Swansea midfielder Gower was replaced by Andrea Orlandi in the 70th minute and the Spaniard's first contribution was to set up Pratley for his side's third goal.
Orlandi broke clear down the left and pulled the ball back from the by-line into the path of Pratley, whose emphatic finish found the top corner from 12 yards.
Swans substitute Lee Trundle fired a shot straight at Steele in the closing stages as the home side comfortably closed out the game.