The Swans delivered yet another strong performance in this clash at Oakwell and did everything but put the ball into the net.
Gorka Pintado, Cedric van der Gun and Nathan Dyer all had good chances in the first-half while the Dutchman had an even better chance early in the second period when he fail to connect properly with the goal gaping.
Craig Beattie was introduced five minutes from the end and the Scotsman had a great chance to win it but fired inches wide of the post on 90 minutes.
The Swans welcomed back Angel Rangel and captain Garry Monk from suspension for this match at Oakwell.
They returned to the defence in place of Marcos Painter and Jazz Richards while Leon Britton and Mark Gower took the places of Jordi Lopez and Joe Allen (hamstring injury) in midfield.
Lee Trundle was also back on the bench and was joined by Tom Butler, who hadn't featured this term due to injury.
Barnsley:David Preece, Rob Kozluk, Stephen Foster, Darren Moore, Julian Gray, Anderson Da Silva, Jacob Butterfield (Nathan Doyle, 59'), Hugo Colace, Adam Hammill (Martin Devaney, 90+2'), Iain Hume, Andy Gray.
Subs: Tom Rusling, Jon Macken, O'Neil Thompson, Onome Sodje, Simon Heslop.
Swans: Dorus de Vries; Angel Rangel, Garry Monk(C), Ashley Williams, Alan Tate; Leon Britton, Andrea Orlandi, Mark Gower (Ferrie Bodde, 68'), Nathan Dyer, Cedric van der Gun (Tom Butler, 75'); Gorka Pintado (Craig Beattie, 86').
Not Used:David Cornell (GK), Lee Trundle, Jordi Lopez, Jazz Richards.
After switching ends to start this contest on a mild afternoon in South Yorkshire it was the home side, under the guidance of new boss Mark Robins, that went straight on the offensive only for the ball to trickle straight through to Dorus de Vries.
Cedric van der Gun, playing on the left of midfield, picked out Nathan Dyer with a pin point cross field pass which ended with the visitors winning a throw-in, but with three minutes on the clock it was the Swans that almost broke the deadlock.
Last season Ashley Williams opened the scoring early on from a set-piece, and it was almost a sense of deja vu as the defender connected with Andrea Orlandi's free-kick but, with the aid of a deflection, saw it dip just wide of the target.
And they came even closer with 11 minutes played as Mark Gower free-kick deep into the area was cleared back into his path, the midfielder broke into the area and rifled his shot inches beyond the fingertips of 'keeper David Preece and just over the top.
The Tykes responded with a chance of their own 60 seconds later as Darren Moore connected with Iain Hume's corner but sent it wide of the target.
But the Swans were enjoying spells of domination and with 14 minutes played created another opening.
This time Garry Monk picked out Andrea Orlandi in midfield, he turned and played a low ball through to Gorka Pintado who was beaten to the ball by Preece, but the former Sunderland stopper's clearance struck the striker and it bounced over the top.
The Swans were showing some nice touches, especially when Leon Britton linked up down the left - the midfielder taking on five Barnsley players before laying off Tate - while Pintado came close with an effort from Orlandi's flick.
But the Tykes, backed by a vociferous home crowd, almost snuck themselves in front on 26 minutes when Andy Gray charged down on goal only for Monk to stop him in his tracks with an excellent, well timed tackle on the six-yard box.
The travelling band of supporters were being treated to some fine football by Paulo's side and with a little over 10 minutes before the break a fine move down the right created another opening.
Angel Rangel, Leon Britton and Nathan Dyer carried the ball forward with neat, one-touch passing, the ball found its way infield to Orlandi whose neat flick found Gower whose low, left-footed strike deflected over for a corner.
Britton saw another shot blocked while a sweeping move from the back ended with Cedric van der Gun cutting inside and sending his effort past the post.
Darren Moore flicked past the post for Barnsley but the dominant Swans had another excellent chance with 41 minutes played - Orlandi's corner was met by Dyer three yards out, he couldn't connect properly and Ashley Williams' effort dropped wide.
Half-Time: Barnsley 0 Swans 0
Within two minutes of the re-start the Swans were handed yet another chance to take the lead; the first effort fell to Nathan Dyer whose strike from 16-yards hit 'keeper Preece, the ball fell to the feet of van der Gun, the Dutchman had the goal open in front of him but as he turned couldn't pull of the shot and the ball was cleared by the defence.
The hosts responded with two set pieces in dangerous areas either side of the box, which were dealt with by the Swans, while Ashley Williams deflected a Mark Gower corner across goal and wide on 52 minutes.
Orlandi curled a left footed shot over the top while Iain Hume beat the offside trap only to shoot high and wide but did have Andy Gray in space to his left.
Mark Robins' side responded well to the pressure they were receiving and looked dangerous when Hume was called on to deliver into the box from set pieces.
The Swans made their first change 22 minutes from the end - Mark Gower was forced off with what looked like an ankle injury and replaced by Ferrie Bodde, playing only his second competitive match since returning from a 10-month period on the sidelines.
Debutant Julian Gray dragged a low shot wide of Dorus' goal before the visitors made a second change - Tom Butler making his return to the first-team to play his first game since May in place of Cedric van der Gun.
Iain Hume was handed another attempt to deliver from a set-piece, this time from more of a central position, but he was well off target with the ball ending up in the stand occupied by the Swansea City supporters.
Craig Beattie was introduced as the final change of the afternoon four minutes from the end, a straight swap for Gorka Pintado, and within minutes of coming on sent a weaker header towards goal from Angel Rangel's cross.
And as the fourth official raised his board to signal three added minutes the Scottish striker had another attempt, this time finding space inside the box only to fire low and wide.