Paul Hayes' 68th-minute goal was enough to seal a 1-0 win for the Iron, but the match's biggest talking point was the retun of Milne.
The Scottish centre-back, 30, has struggled with a knee injury but came on at half-time to replace Rob Jones, who had a hamstring problem.
"I'm delighted for Kenny. He has gone through in the last 17 months what no player should have to," said Adkins.
"He's had two operations on his knee and worked so hard to get himself back in contention.
"The win is 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."
Barnsley's best goalscoring chance fell to Daniel Bogdanovic, who could not find the target. Mark Robins, the Tykes, manager, said his side had performed poorly.
"We didn't reach the standards which we have been setting ourselves in recent games," he said.
"This was a competition I really wanted to go further in, I'm disappointed, we never really seemed to get going."