Bacuna's superb 76th -minute free-kick was Villa's first goal in five matches and Kozak’s cushioned header eight minutes later proved enough to earn Paul Lambert’s side their first win since the 3-2 victory over Manchester City at the end of September.
The closest the visitors came to scoring was in the 10th minute as Kozak almost put through his own net, but otherwise Malky Mackay’s side created very little in the way of goalscoring opportunities, failing to build on their memorable 1-0 victory over Welsh rivals Swansea City last week.
Manager Paul Lambert made two changes from last week's draw at West Ham, Aleksandar Tonev and Kozak replacing Matt Lowton and Andreas Weimann, who was absent with a hamstring injury picked up in the Upton Park clash.
For Cardiff, there was just one alteration from Sunday’s derby triumph over Swansea as Aron Gunnarsson came in for Jordon Mutch, who dropped to the bench.
The hosts started the brighter, Marshall stretching to parry Karim El Ahmadi's effort from distance in the fifth minute, Steven Caulker nipping the ball away from Christian Benteke as the Belgium international looked to convert the rebound.
Kozak almost gifted Cardiff the lead five minutes later, the striker inexplicably nodding Gary Medel's free-kick narrowly wide of his own goal under no pressure.
In a frantic opening, Brad Guzan rushed from his line to deny Gunnarsson in the 12th minute before the pace of the game slowed noticeably, the American comfortably catching Craig Bellamy’s curling free-kick on the half-hour mark with both sides struggling to create openings.
Goalmouth action was equally scarce after the break, with Ashley Westood fizzing a 25-yard shot narrowly wide in the 63rd minute as each struggled to make any impression on the opposition defences.
However, the hosts made a crucial breakthrough 13 minutes later, Bacuna bending the ball brilliantly around the wall from 25 yards to the relief of the home support inside Villa Park.
Shortly afterwards, Villa doubled the lead, Lowton nodding the ball back across goal where Kozak was on hand to redirect beyond a despairing Marshall to seal the points and send Villa above their opponents in the table.
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