Wilfried Gnonto and Patrick Bamford both scored braces, while top scorer Rodrigo also got his name on the scoresheet before Callum Robinson netted two consolation strikes for the Bluebirds in the closing stages.
Leeds had been eliminated from the FA Cup third round in each of the last five seasons, but that run has now come to an end as they have set up a favourable fourth-round tie away against either Accrington Stanley or Boreham Wood, which will take place later this month.
Just 10 days ago, Leeds were frustrated by Cardiff in the reverse fixture as they were required to rescue a 2-2 draw against their Championship counterparts with two goals in the final half-hour.
However, Jesse Marsch's men had no problems making a fast start in the replay on home soil, with Gnonto finding the net with a stunning acrobatic volley in the first minute after receiving a perfectly weighted delivery from Rodrigo.
Leeds then netted twice within a three-minute period midway through the first half, with Rodrigo doubling the hosts' advantage with a confident, composed left-footed strike, taking his tally for the season to 12 in all competitions.
Gnonto then fired a right-footed shot past Jak Alnwick at his near post following a well-worked counter-attacking move to put Leeds three goals to the good in the 36th minute.
Cardiff, who had caretaker boss Dean Whitehead in the dugout after Mark Hudson was sacked on Saturday, thought they had pulled one back on the stroke of half time, but Andy Rinomhota was deemed to have been in an offside position before Curtis Nelson headed home from close range.
Bamford, Luke Ayling and Brenden Aaronson all came on early in the second half, and it was the former two who combined for Leeds' fourth goal in the 71st minute.
From inside his own half, Ayling played a curling ball over the top of Cardiff's backline, with Bamford bending his run from the right channel before placing a side-footed finish into the far corner.
Bamford then doubled his tally just five minutes later, receiving a through-ball from Joe Gelhardt before directing a strike with the outside of his left boot into the same corner.
Cardiff fans were able to cheer a couple of consolations strikes within the closing stages of the match, with Robinson capitalising on a defensive mishap from Ayling to pull one back with a header inside the six-yard box in the 83rd minute.
Robinson then converted from the penalty spot in the third minute of stoppage time after VAR spotted Marc Roca handle the ball inside the area.
Leeds will now turn their attention back to the Premier League with a home encounter against Brentford on Sunday, while managerless Cardiff welcome Millwall to South Wales in the Championship on Saturday.