The game ended 2-2 after extra-time, before misses from Ade Akinfenwa and Alan Tate proved disastrous for the Swans, who were looking for a second straight promotion.
After City made the brighter start, it was the Tykes who took the lead 19 minutes in as Martin Devaney's cross in from the right was flicked on and Hayes was given time at the far post to hammer home a low volley.
The Swans drew level nine minutes later as former Barnsley striker Rory Fallon executed a superb overhead kick with his left foot after Owain Tudur-Jones had nodded on a free-kick into the area.
Five minutes before the interval Swansea were gifted the lead as Tykes keeper Nick Colgan allowed Andy Robinson 25-yard shot to slip from his grasp and trickle over the goal-line.
The second half started in a much more sedate fashion, before Barnsley made it 2-2 just past the hour mark.
Brian Howard was hacked down just outside the area and Danny Nardiello curled his right-foot free-kick past the unsighted Willy Gueret.
Tykes skipper Paul Reid almost snatched a dramatic late winner as his header from Devaney's superb cross bounced up and over the crossbar, while Tate wasted a chance right at the death - slicing wide of target from ten yards out.
Swans talisman Lee Trundle forced a fine save from Colgan early on in extra-time, but the City striker then wasted a clear chance when placing a shot wide of target from close in midway through the second period of extra-time.
That left spot-kicks to decide the winners, with Barnsley scoring all four of their kicks, through Hayes, Chris Shuker, Paul Heckingbottom and Antony Kay to emerge as 4-3 victors.