Keyrol Figueroa scored twice and Lucas Pitt added another in the second half for the Reds, who found themselves 3-0 down at the break.
Thierry Katsukunya and Caden Voice both headed in from corners to put Cardiff in a commanding lead inside the opening eight minutes.
The Welsh visitors then netted a third before half-time with a long-range strike from Cody Twose.
Figueroa's clinical finish reduced the deficit in the 55th minute, with Isaac Mabaya and Wellity Lucky later coming on to make their returns from injury.
Jake Davies restored Cardiff's three-goal advantage but it was quickly down to one as a brilliant Figueroa volley was followed by a header from Pitt.
An equaliser was nearly found right at the end, though no player in red was able to get on the end of Mabaya's teasing cross.
