They started brightly, though, and midfielder Karim El Ahmadi's long-range shot was parried by Cardiff goalkeeper David Marshall early on with striker Christian Benteke unable to react quickly enough to latch on to the rebound.
Benteke's strike partner Kozak went close to putting through his own net at the other end moments later when his header went inches past the post.
Steve WilsonBBC Match of the Day commentator
"After 15 minutes you would have banked on it being a thriller at the Villa, but as the game wore on it looks more like it would be a nil-nil-er. Paul Lambert's side went seven-and-half hours without a goal but always looked the more likely team of the two teams to score in the second half. And the long wait for a goal was finally brought to an end in style by Leandro Bacuna before Libor Kozak headed home Villa's second."
A defensive mix-up let Cardiff midfielder Aron Gunnarsson pinch the ball on the edge of the penalty area in the 12th minute, but Villa goalkeeper Brad Guzan scampered off his line to snuff out the danger.
The remainder of the first half became scrappy with chances at a premium.
joined Villa from Lazio for £7m
on deadline day, brought a fine diving save out of Marshall moments after the half-time interval when he got his head on the end of Bacuna's cross from the right.
The hosts looked the more likely side to break the deadlock and almost caught Cardiff on the counter attack when El Ahmadi crossed for Benteke but the Belgium international directed his header wide.
Leandro Bacuna goal against Cardiff was fabulous - Aston Villa's Paul Lambert
The 21-year-old winger dusted himself down and curled the resulting set-piece around the Cardiff wall and into the net with Marshall rooted to the spot.
Koazk made the points safe late on when he scored Villa's second with a close-range header after Matthew Lowton had nodded the ball back across goal.
Cardiff pressed to get themselves back into the game in the remaining minutes but the home defence stood firm to secure the points.
Aston Villa manager Paul Lambert:
"The players battled and were very good. Leandro's free-kick was fabulous. His goal against Manchester City was brilliant but I think that surpasses it.
"I thought we deserved to win the game. I knew it would be a tough game so it's a good win.
"The clean sheet gives us confidence going forward and I was really pleased with the result. The lads were excellent."
Cardiff City manager Malky Mckay:
"Ultimately it was a good goal that made the difference. I thought we had played well until then. We had plenty of possession and were comfortable until that point. It was a poor free-kick to give away.
"Top players at this level can put the ball in the top corner and Leandro Bacuna punished us by doing that. We will learn out lesson from that.
"Overall, I'm not too disappointed because if we show that kind of application we'll be fine."
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