In a roller-coaster match that saw the Premier League leaders twice fall behind only to grab a dramatic 122nd-minute equaliser, it was left to teenager Gael Kakuta to fluff the decisive penalty kick.
Big-spending Manchester City beat a young Arsenal team 3-0 in the night's other quarter-final to set up a mouth-watering clash with holders and local rivals Manchester United in the last four.
Blackburn went back in front in extra time with a Benni McCarthy penalty before Robinson flapped at a cross from the left and Paulo Ferreira grabbed Chelsea's third goal in time added on.
Kalinic and Chelsea's Michael Ballack both missed in the shootout before Robinson saved a weak attempt from Kakuta.
"We are disappointed but we have to look at the match, I think that the first half was not good but the second and extra time was very good," Ancelotti told Sky Sports television.
"We played 10 against 11 and showed a very good mental attitude and spirit."
Blackburn's assistant manager Neil McDonald praised his players' spirit: "You always have to have the belief," he said.
"We always felt that if they changed the team we could get at them and try and stop the flow of football."
