Joe Ledley and Michael Chopra were both on target before the break to put the home side firmly in control in Tuesday night's Championship clash at Ninian Park.
Barnsley did hit back after the break when Jon Macken scored after 62 minutes, but their hopes of a comeback were undermined three minutes later when Andranik Teymourian was given his marching orders.
Peter Whittingham ensured victory for Cardiff two minutes from the end as the Bluebirds struck three goals for the first time this season.
"We have now shown that there is healthy competition for places and today we were awesome in the first half and could have come in four goals up," Jones said.
"In the second half, we took our foot off the gas a little and for 10 minutes or so, they gave us a few anxious moments and it was a poor goal to give away.
"We still controlled the game after that and created a number of chances but unfortunately we have picked up a few more knocks."
The Bluebirds took a fourth-minute lead when the Ledley was left free to volley home from a Whittingham corner.
Such was the Cardiff dominance that keeper Dimi Konstantopoulos was not called upon to make a save until the 41st minute when he leapt to tip over a deflected Michael Mifsud shot.
Two minutes later, City had doubled their lead when a Whittingham long-range effort was spilled by Heinz Muller in the visitor's goal and Chopra nipped in to fire back across the keeper and into the net.
Macken, who had only been on the field one minute, reduced the deficit with his first touch on the ball in the 62nd minute when Roger Johnson failed to clear a towering Darren Moore header.
There was a flare up three minutes later and Teymourian was given a straight red for an alleged slap on Chopra after the City striker had won a free-kick on the edge of the area.
City then controlled possession with their extra man, but it was not until the last minute that they put the game beyond the visitors.
Eddie Johnson switched the ball wide to Whittingham who cut in before steering a superb curler beyond Muller and in off the far post.
Barnsley manager Simon Davey blasted referee Pat Miller for his performance after the game.
The Tykes boss was sent to the stands by Miller at half time - a decision he was still fuming about after the final whistle.
"It's frustrating because I thought that the referee wasn't up to standard," Davey said.
"I can't say much but he told me I couldn't watch the match in the second half.
"When there is a hostile crowd some people can get overawed and influenced for decisions they have to make for the home fans."