While the Yorkshire side defended strongly around the box, they were undone by a penalty and two perfect passes from the midfield.
The first midway through the opening half came from Australian Luke Wilkshire.
Seeing a gap in the back line between Robbie Williams and Bobby Hassell, he pushed a well measured pass through to Steve Brooker. He took it in his stride to drive right footed from 20 yards inside the left-hand post.
If that was bad for Barnsley the next two strikes were disastrous.
Chasing a ball into the left side of the box Marcus Stewart was upended by Dale Tonge.
There was no doubt about the penalty and Stewart, with a clever five pace run, caught Nick Colgan off balance as he converted it.
Six minutes later the former Sunderland finisher, who has found it hard to strike form in League One, had the ball in the net again.
This one was set up by Scott Brown, who robbed Tonge near the touchline to cut in and release to Stewart unmarked on the far side of goal.
The scoreline was harsh but the script looked predetermined and City even had luck early in the second half when there was a strong penalty call for handball against midfielder Cole Skuse.
City's new manager had made an emergency loan, taking 20-year-old Osei Sankofa from Charlton for a month, but the deal went through so late that the new centre-back only met his new team-mates in the dressing room an hour before kick-off.
He was given an early examination by Barnsley's Paul Hayes but quickly settled into the back four.
For Barnsley, Martin Devaney and Chris Shuker worked the flanks well, but as a team The Tykes, in the middle of a poor run, never quite created enough pressure in the final third.