Bluebirds manager Dave Jones made 10 changes from the side which started the 1-0 win at Leeds on Saturday but was made to pay as Kandol scored once in each half for Paul Fairclough's Bees to progress to the second round.
Cardiff started well and Roger Johnson twice headed narrowly off target in the first quarter of an hour before debutant goalkeeper Mark Howard had a moment to forget.
The former Arsenal trainee dashed out of his area to clear but his mis-hit clearance fell straight to Kandol on the halfway line and he chipped home into the empty net.
And Barnet doubled their lead eight minutes into the second period when debutant Joe Devera whipped a cross in from the right and Kandol climbed highest to head past Howard.
Jones threw on Wales internationals Paul Parry and Joe Ledley as the home side tried to mount a comeback but although Darcy Blake hit the post from 30 yards out late on, the Bees side held on for a memorable win.