"This was a competition I really wanted to go further in," said Robins. "I'm disappointed, we never really seemed to get going.
"We didn't reach the standards which we have been setting ourselves in recent games.
"I don't really know why, perhaps it was over-confidence from the players.
"When you get that this game has a habit of kicking you in the teeth and that's what's happened to us.
"We had enough chances to have at least earned a replay but we did not work hard enough to get the result we wanted."
Barnsley were sunk by a goal after 68 minutes from their old boy Hayes.
He pounced at the far post to convert a cross from Jonathan Forte for his seventh goal of the season.
Earlier, Scunthorpe had missed a couple of good chances when first Garry Thompson and then Forte failed to find the target.
Barnsley, too, missed a golden opportunity to get back on level terms when striker Daniel Bogdanovic finished wastefully after working himself a good opening.
It was a perfect return to football for Scunthorpe's centre-back Kenny Milne, who came on at half-time for his first game in 17 months after recovering from a career-threatening knee injury.
"I'm delighted for Kenny," said manager Nigel Adkins. "He has gone through in the last 17 months what no player should have to.
"He's had two operations on his knee and worked so hard to get himself back in contention.
"He was on the bench because of a number of injuries which we have.
"For us, it's a great start to a new year and a great start to a new decade.
"We wanted to progress in this competition and now we're hoping for a big draw in the fourth round."
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