Robinson saved from Michael Ballack and Gael Kakuta, who took the last kick.
Benni McCarthy, Brett Emerton, Vince Grella and David Hoilett scored Rovers’ penalties, with Nikola Kalinic having his saved. Didier Drogba, Florent Malouda and Yury Zhirkov were successful for Chelsea.
In normal time, Blackburn had taken an early lead through Kalinic but Chelsea hit back through Drogba and Salomon Kalou, before Emerton levelled.
But Rovers went ahead again in extra-time through a McCarthy penalty, only for Paulo Ferreira equalise in the final seconds.
Rovers will now meet Aston Villa in a two-legged semi-final, the first leg taking place during the week commencing Jan 4 with the return a fortnight later.