Robinson saved efforts from Michael Ballack and Gael Kakuta as Rovers teed up a semi-final showdown with Aston Villa.
It was a fitting climax to a gripping night of drama at Ewood Park, which saw Rovers twice take the lead only for Chelsea to come roaring back.
Rovers lined up with striker Benni McCarthy playing at the tip of a midfield diamond and Nikola Kalinic partnering Jason Roberts in attack.
Chelsea boss Carlo Ancelotti rested Petr Cech, Ashley Cole, John Terry, Michael Essien, Frank Lampard and Nicolas Anelka, while leading scorer Didier Drogba started on the bench.
The absence of so many leading lights from the Chelsea teamsheet clearly spurred on the home side and they looked up for the challenge from the off.
Kalinic fired an early warning when he shot wide following a neat exchange with Roberts.
However, Chelsea's reprieve proved shortlived as the Croat promptly fired Rovers ahead in the ninth minute.
Roberts combined with Morten Gamst Pedersen on the edge of the penalty area, before the latter slipped in Pascal Chimbonda on the left.
The Frenchman then darted towards the byline and whipped in a low cross, which was gleefully turned home from six yards out by Kalinic.
Rovers had further chances to extend their lead with Brett Emerton volleying narrowly wide from distance and then Kalinic lost his footing as he bore down dangerously on goal.
In contrast, little was seen of Chelsea as an attacking force, apart from a Salomon Kalou effort which sailed harmlessly wide.
Carlo Ancelotti knew he had to make changes and a triple substitution at the beginning of the second half turned the tie on its head.
It took Drogba just two minutes to make his mark with the Ivorian out-jumping Ryan Nelsen to plant a powerful header past Robinson.
Suddenly the momentum was with Chelsea and they made it 2-1 in the 52nd minute with Kalou racing onto a through-ball from Yury Zhirkov before slotting coolly past Robinson.
However, Rovers refused to lie down and they levelled the scores again 12 minutes later.
Emerton's cross from the right took a wicked deflection and as Henrique Hilario came to claim it, he appeared to be distracted by Kalinic's presence at the near post, leaving the ball to sail into the net.
Kalinic had a glorious chance to win it in the dying seconds of normal time but was denied by Hilario.
So to extra time and within two minutes of the restart Rovers were awarded a penalty when Zhirkov fouled David Hoilett.
McCarthy took responsibility and confidently dispatched his spot-kick past Hilario.
However, Chelsea fought back again and a mistake from Robinson in the final seconds of extra-time presented Ferreira with an equaliser the Londoners barely deserved.
But the Rovers keeper then atoned for his error in the shootout with two saves to deny Ballack and Kakuta.