Portsmouth 2 Plymouth Argyle 1
Kranjcar struck the decisive goal in the final minute of the first half to keep alive Pompey manager Harry Redknapp's hopes of landing the first major trophy of his 25 years in the dugout.
Kranjcar's close-range effort came 11 minutes after midfielder Lassana Diarra had cancelled out Chris Clark's early opener for the Pilgrims.
But the Premier League European hopefuls had goalkeeper David James to thank for securing their passage into the fifth round.
Plymouth, some 20 places below Pompey in the league pyramid, were gifted a surprise fifth-minute lead when Clark's cross-cum-shot was deflected in off the outstretched right boot of a despairing Hermann Hreidarsson.
Plymouth would have doubled their lead two minutes later but for a superb save from James to keep out Rory Fallon's shot on the turn.
It took until the 33rd minute for Portsmouth to register their first shot on target, but it was worth the wait.
Pedro Mendes took a short corner on the right and the ball broke free to Diarra, who curled an unstoppable shot into the bottom left corner from 25 yards.
David Nugent, hero of Portsmouth's third-round win over Ipswich, should have put the home side in front four minutes later when he found himself one-on-one with Luke McCormick, but scuffed his shot into the arms of the goalkeeper.
Jermaine Easter came within inches of restoring Championship Plymouth's lead two minutes before half-time when he fired against a post from a tight angle.
Easter's misery was compounded when Kranjcar turned the ball home from close range just a minute later, after latching on to a Glen Johnson cross.
James was by far the busier of the two keepers after the break and produced a brilliant 53rd minute save when he flung himself across the goal-line to somehow stop Easter's effort from nestling in the top corner.
Portsmouth's 12-goal top-scorer Benjani almost made the game safe when he fired just wide of McCormick's right post late on, but the miss did not come back to haunt Redknapp as the home side progressed.