The beleaguered boss, looking for some respite from his team's awful form in the Barclays Premier League, found only more misery as Eden Hazard's spot-kick was saved by Jack Butland following a 1-1 draw, giving Stoke a 5-4 shoot-out success.
The defeat was Chelsea's fourth in their last six games in all competitions and will turn up the heat yet further on the under-fire Portuguese manager.
The holders looked set to lose in 90 minutes as Jonathan Walters' 52nd-minute goal, a fine shot on the turn from the edge of the area, put the Potters in front.
But Loic Remy, on for the injured Diego Costa before half-time, fired home in injury-time when the ball broke to him following a corner, and Stoke lost Phil Bardsley to a second yellow card shortly after.
But the hosts held on for penalties and after nine successful attempts Butland denied Hazard to send Stoke into the quarter-finals.
The Gunners lost Alex Oxlade-Chamberlain and his replacement Theo Walcott to injury inside the opening 20 minutes and were two goals down at half-time thanks to Ross Wallace's sharp finish in the 27th minute and Lucas Joao's towering header in the 40th.
The visitors' hopes of a second-half fightback were dashed six minutes after the restart when Sam Hutchinson grabbed a close-range third.
Out-of-form Norwich threatened an unexpected result too when Sebastien Bassong fired them into a 51st-minute lead at Everton, but Leon Osman equalised 17 minutes later.
It was 1-1 at the end of extra-time, Everton winning 4-3 on penalties, with Wes Hoolahan and Nathan Redmond missing for Norwich.
Hull were also successful on penalties, knocking out Leicester 5-4 after a 1-1 draw.
Riyad Mahrez failed with the first attempt of the shoot-out, having earlier put the Foxes in front 10 minutes into extra-time - a goal that was soon matched by Hull's Abel Hernandez.
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