The garish camouflage kit worn by Everton's goalkeepers does not disguise their embarrassment. Jan Mucha, their Slovakian stand-in, was at fault for both Chelsea goals as André Villas-Boas's team survived a third red card in two matches to advance into the quarter-finals. The Chelsea manager reacted in a Mourinho-esque manner to the final whistle, leaping into the air and leading his players on a shirt-throwing, chest-thumping celebration before the away hordes. Here the Carling Cup became a test of inner resources.

Ross Turnbull was dismissed from the Chelsea goal for a 58th minute-foul on Louis Saha, enabling Petr Cech to enter from the bench and save the resulting spot-kick from Leighton Baines. The penalty misses and red cards were ultimately shared, Nicolas Anelka also failing for the visitors and Royston Drenthe collecting a late red card that prompted Chelsea to seize control of extra time.

Everton had the chances to have won against 10 men in normal time but there was no denying Chelsea's superiority on a level playing field. When Mucha could only parry Florent Malouda's shot straight to Daniel Sturridge, having earlier handed Chelsea the lead with an amateurish mistake, the substitute finished expertly and a draining victory was secured.

"We were very committed, we showed strength of character and resolve once again," said Villas-Boas who, unlike at Queens Park Rangers on Sunday, had no complaints over the red card. "Having 10 men unfortunately made it difficult for us again but to survive in difficult circumstances is an extremely good sign for us. It shows the squad is committed to all the trophies. It was important to win after a defeat that happens like it did at Queens Park Rangers. To do it in this fashion is more gratifying. The emotional victory you get from a result like this is immense."

There was no place in the Chelsea squad for John Terry and no questions allowed about the England captain either as a consequence of the Football Association's investigation into allegations of racist abuse against Anton Ferdinand, which he denies. Villas-Boas made 10 changes in all but there was minimal disruption to the visitors' play and they should have established an early lead from the spot.

John Heitinga, recalled to the heart of the home defence but rusty on his first start in five matches, ploughed through Josh McEachran as the young Chelsea midfielder gathered the rebound from his own shot against Sylvain Distin. Heitinga still had the nerve to plead innocence but was spared when Anelka ambled forward, sent Mucha the wrong way but paid for his casual approach as the ball drifted well wide.

Moments later Anelka was bundled off the ball inside the area by Distin only for the referee Lee Mason to wave away appeals for a second spot-kick. Just as Everton began to settle, however, with Drenthe increasingly influential, they threw it away – literally. There appeared little danger when Salomon Kalou floated a first-time chip towards the Everton goal from the edge of the area. The Slovakia international Mucha, given a rare outing in place of the rested Tim Howard, came to collect with both hands but made a complete hash of the catch and let it slip through his grasp and over the line. Moyes turned towards his own bench open-mouthed. Even Kalou had the decency to look embarrassed.

"I don't think it had an effect on the players after that," said Everton's manager, David Moyes. "The boys played really well, especially in the second half and we were unlucky not to win in 90 minutes but we've only got ourselves to blame for not winning it."

The introduction of Seamus Coleman gave Everton much-needed speed and width and with seven minutes remaining he delivered an inviting cross on to the head of Saha, whose glancing finish broke Cech's impressive resistance. Saha almost struck again in the final seconds of normal time but his low shot flashed narrowly wide. Chelsea, despite their disadvantage, also continued to press for a winner with Sturridge introduced and David Luiz testinge Mucha with a dipping free-kick.

Drenthe received the second red card of the game when he collected a second booking for a foul on Ryan Bertrand in extra time and a suspension that rules him out of Manchester United's visit to Goodison on Saturday. Anelka then struck a post following a fine run and cross from Bertrand and Mucha produced a fine save to deny Branislav Ivanovic from distance. His next stop merely proved the assist for Chelsea's winner.