Paul Caddis' brace - his first Birmingham goals - put the visitors in control before Nikola Zigic sealed the three points eight minutes shy of half-time.
Full-back Caddis lashed a 25-yard shot into the bottom right-hand corner in the 13th minute and doubled his side's lead with a dubious penalty after Zigic went down under Jean-Yves M'Voto's challenge nine minutes later.
Zigic sealed the points with a perfectly executed chip over Jack Butland, following an error by Tykes midfielder Jim O'Brien.
Clark said it would be disrespectful to call the win comfortable but was happy with his team's display.
He said: "Our whole performance in the first half was very good but we knew Barnsley would regroup and come at us a little bit and take a few gambles and risks so we had to defend very, very well in the second half and we did that.
"I'm delighted that we got a clean sheet on top of three excellent goals.
"I'd had a look all week at the build-up to the game and what was being said from this (Barnsley's) side of the camp and I sensed that they were talking about it being a must-win and a pressure game, and a crunch match.
"I just wanted our players to come and express themselves. We've got a very, very young group and they did that.
"We handled the ball well, we caused problems and it caused tension around the stadium, which transferred itself on to the pitch a little bit and that plays into the away team's hands.
"We've been quite impressive away from home in the last few weeks, taken seven points from nine; tough games at Derby, Huddersfield and here so we're delighted with that return."
Clark, referring to the penalty incident, added: "The referee seemed to be in a good position.
"We've been on the end of some strange decisions this season and we've had to dust ourselves down, I'm sure David (Flitcroft) is frustrated with that. He did have a conversation as Nikola (Zigic) came off and he asked him hand on heart did he think it was a penalty and Nikola replied that he did."
Barnsley manager David Flitcroft did not attend the post-match press conference.