Clark was indebted to his skipper Steven Caldwell as the Scot scored his first goal for Birmingham to give them a 2-1 lead after an indifferent start when Barnet were often the better side.
Jon Nurse gave Barnet a 31st-minute lead which was cancelled out by a Marlon King penalty eight minutes later.
Caldwell then knocked the fight out of Barnet with his 49th-minute goal which was followed by Darren Ambrose who notched a third in the 54th minute.
Peter Lovenkrands scored the fourth three minutes from time and Wade Elliott scored a fifth a minute later.
Clark said: "We didn't cause them enough problems early on.
"We had to change a few things at half-time. The players and myself were in agreement what was necessary and we did what was required.
"The second half performance was a big improvement.Not only did we score four goals we actually didn't give them much sight of our goal.
"Basically, it was not us playing at the right speed. Certainly when we did it in the second half that we looked a different side.
"I have to praise the players. They are a honest bunch of players and they are top players who have high demands of themselves."
Barnet manager Mark Robson, in charge for the first time since his arrival from Peterborough, admitted: "I was absolutely delighted with the way my team played.
"We were excellent and to be honest we could have gone in 3-1 at half-time instead of 1-1.
"Even in the second half we were quite good. The early goal didn't help and then Darren Ambrose scored a fantastic goal.
That is the class Birmingham have. Their finishing was first class.
"Even at 3-1 I could have taken it on the chin. The only disappointment I had was the last 10 minutes.
"We caused our own problems and finished up losing 5-1 which was hardly a fair reflection on the game.
"There were lots of positives for me to take out of the game."