The home side - beaten in their previous two matches - had Kaladze dismissed 11 minutes from time but held on for a point against Eusebio Di Francesco's Lecce, who had lost their last three.
Genoa started brightly and came close after five minutes when Juraj Kucka set up Alexander Merkel - but goalkeeper Massimiliano Benassi kept out his effort.
Two minutes later Lecce's Djamel Mesbah shot narrowly over from long range, before ending a mazy run with a shot just wide in the 15th minute.
Benassi saved Marco Rossi's 19th-minute header, then tipped over Kucka's left foot cross-shot on the half hour.
The Lecce keeper was busy again when he made a great diving save to keep out Miguel Veloso's 32nd-minute free-kick.
Two minutes after the restart Lecce needed defender Nenad Tomovic to get in the way of a goalbound effort from Rodrigo Palacio, who had been set up by Andrea Caracciolo.
However, the visitors threatened four minutes later when Cristian Pasquato skimmed the Genoa goal with a curling shot, following excellent work by Guillermo Giacomazzi and Carlos Grossmuller.
Benassi denied Kucka in the 55th minute, then four minutes later the Slovakian was off target with another effort.
The keeper was beaten in the 64th minute but Genoa substitute Cristobal Jorquera's shot crashed back off the post and was scrambled to safety before Palacio could capitalise.
However, the home side's momentum was checked when Georgian international defender Kaladze was booked for a foul - and received another yellow card for dissent to end his participation in the game.
Genoa came closest to a goal in the closing stages despite being a man down - but Benassi dived to his right to keep out Bosko Jankovic's powerful free-kick four minutes into injury-time.