Having knocked out Ipswich Town of League One in the previous round, the Bluebirds may have been optimistic ahead of the first competitive meeting between these two sides for 52 years.
The Championship club had the first chance of the game when Callum Brittain tested Paul Farman from range but the keeper did well to tip his effort over the bar.
Barnsley opened the scoring in the 23rd minute after Callum Styles whipped in a brilliant free-kick and captain Mads Andersen leapt the highest to head past Barrow’s number one.
Finding an equaliser became a much tougher ask when Barrow striker Tom Beadling was sent off for a high challenge on Romal Palmer after 38 minutes.
Things went from bad to worse for the fourth tier side after 42 minutes when Jordan Williams mishit his first attempt but made no mistake from the second to give the Tykes a comfortable lead going into half-time.
Remy Vita could have added a third after the break when he latched on to Jack Walton’s punt forward but the Barnsley man failed to capitalise.
Ollie Banks then scored a goal of the round contender in the 61st minute with a thunderous free-kick from the Barrow skipper to reduce the deficit.
The first equaliser of the night came after 78 minutes when substitute Luke James found Anthony Glennon in the box and the debutant finished confidently to make it 2-2.
Barrow’s joy was short-lived as striker Devante Cole curled in from distance after 83 minutes to give the Tykes the advantage once more.
The Bluebirds refused to be beaten and three minutes later, Remeao Hutton’s cross was met by James Jones who headed powerfully past Walton to level the scores again.
Substitute Carlton Morris must have thought he had won it for Barnsley when he scored in the 88th minute.
Unbelievably, Josh Kay, another man to come off the bench, drilled in from distance in the 90th minute to take this amazing tie into a further 30-minute period.
Morris got his second of the evening 12 minutes into extra time and that goal proved to be enough to see Barnsley through to the fourth round but Barrow will go home very proud of their performance.