First-half strikes from Jaden Philogene-Bidace and Sheyi Ojo put the Bluebirds in control at the break, but the Whites fought back in the final half hour with Rodrigo pulling one back in the 65th minute before fellow substitute Sonny Perkins netted a dramatic 93rd-minute equaliser.
Both teams are set to lock horns in a third-round replay which is scheduled to take place on either January 17 or 18, and the winners will face either Boreham Wood or Accrington Stanley in the fourth round.
Leeds head coach Jesse Marsch, who named a starting lineup with an average age of just over 24, will have been relatively pleased with his side's start to proceedings, dominating possession and preventing the hosts from carving out any clear-cut opportunities.
However, Cardiff - the lowest scorers in the Championship so far this season - opened the scoring against the run of play courtesy of a close-range strike from Philogene-Bidace in the 24th minute.
Pascal Struijk was unaware of Mark Harris's well-timed run in behind, and the Bluebirds attacker forced Joel Robles to make a save at an acute angle before the ball eventually deflected its way to Philogene-Bidace to place a shot beyond Rasmus Kristensen on the line.
Cardiff's opener helped them to grow into the game and it went from bad to worse for the visitors just seven minutes later, when Sheyi Ojo fired a right-footed strike into the roof of the net after controlling a floated delivery over the top of Leeds' backline from Andy Rinomhota.
Leeds were sloppy in possession early in the second half and Marsch reacted by bringing on former Cardiff loanee Cody Drameh, top scorer Rodrigo and January signing Maximilian Wober to provide a much-need spark, the latter of whom - naturally a defender - operated in centre-midfield.
The Whites should have pulled one back just after the hour mark when Struijk directed a header narrowly wide of the far post following a Sam Greenwood corner.
However, Leeds made amends just moments later when Rodrigo met Greenwood's cross from the left and thumped a header beyond Cardiff goalkeeper Jak Alnwick to restore hope for the travelling away supporters.
Leeds were then awarded a penalty 10 minutes from time when Cardiff's Joel Bagan used his arms to block a shot from Junior Firpo, and the defender was subsequently sent off for his actions.
Rodrigo stepped up from 12 yards but saw his tame effort saved low down by Alnwick to the delight of the Cardiff supporters.
The home side's celebrations were short-lived, however, as Leeds managed to salvage a draw in the third minute of stoppage time when Firpo flicked a Wilfried Gnonto cross towards Perkins inside the six-yard box and the 18-year-old netted his first senior goal to force a replay.
Leeds will now turn their attention to Friday's Premier League trip to Aston Villa, while Cardiff will take on Wigan Athletic in the Championship on Saturday.