The Tykes are now unbeaten in their last five matches after goalkeeper Samuel Radlinger made a string of crucial saves to keep the scores level.
They are now level on 21 points with Stoke in the battle to stay in the second tier of English football.
Swansea moved a point away from the play-off places despite failing to win for the second game in a row.
Boss Steve Cooper made two changes to his Swans side following their Boxing Day defeat at Brentford.
Kyle Naughton came in at right-back in place of Connor Roberts and Jay Fulton replaced George Byers in the centre of midfield.
Barnsley’s eight-point haul from five festive fixtures had given them fresh hope of survival.
Manager Gerhard Struber stuck with the XI that drew with Championship leaders West Brom after Alex Mowatt recovered from illness.
The Tykes midfielder saw yellow in the 22nd minute for tugging Andre Ayew’s shirt after an uneventful opening quarter.
Barnsley thought they had snatched the lead when Conor Chaplin flicked Cauley Woodrow’s back-heel past Freddie Woodman, but the goal was ruled out for offside.
Radlinger then made two important saves to keep the visitors on level terms. He got down sharply to deny Bersant Celina after he was picked out by Ayew before using his lightning reactions to deny Sam Surridge’s close-range shot at his back post.
The Tykes goalkeeper was almost caught out 10 minutes before half-time after racing out of his area to head a loose ball clear.
Surridge sent a dipping shot over the head of Radlinger from 30 yards after his header fell at the feet of Tom Carroll.
The goalkeeper scrambled back as the ball looped over his head, but he was able to breathe a sigh of relief as the ball rattled the crossbar.
Swansea were thanking the woodwork after half-time when Mowatt’s cross found Jacob Brown, but the striker slammed a shot against a post from three yards.
Cooper’s side had another clear-cut opportunity to break the deadlock in the 56th minute when Ayew headed Matt Grimes’ corner towards goal, but Radlinger was there once again to scramble the ball away with his feet.
Substitute Byers was next to go close when he sent a first-time shot half-a-yard wide after being teed-up by Naughton’s low cross.
But as the clock ticked down towards the final whistle, both sides seemed content with a point.