First-half strikes from Jaden Philogene 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 will both be in the hat for the fourth-round draw later today.
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 in the 24th minute.
Patrick 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 to place a shot beyond Rasmussen 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 head 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.
More to follow.