Eden Hazard, symbol of Chelsea's early-season woes, was the fall guy in Tuesday's fourth-round tie at the Britannia Stadium.
Stoke goalkeeper Jack Butland saved the Belgian winger's spot-kick to give the hosts a 5-4 shootout win after the game had finished 1-1 following extra time.
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With Chelsea 15th in the Premier League and out of the League Cup, both of which they won last season, Mourinho faces mounting speculation that he could be sacked, but ahead of Saturday's home game with Liverpool, he said that he was feeling calm.
"I have a day off tomorrow, a fantastic family, I can sleep well every night," said Mourinho, whose side have already succumbed to eight defeats in all competitions, including the Community Shield.
"I'm going to enjoy my day and Thursday, it will be one more day like I have had in the last 15 years of my life -- honest and dedicated."
Chelsea's hierarchy gave Mourinho a public vote of confidence following the recent home 3-1 defeat by Southampton, but he said that he did not need more reassurances about his position.
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"I don't need it. I don't need more," he told his post-match press conference.
He also hit out at newspaper reports that he has lost the support of his players.
"Honestly, do you think they are not with me? That they didn't give everything to win the game?" he said.
"That is really sad -- not sad for me, but sad for the players -- and I think it is a lack of respect to the players, not to me.
"For me it would be a fantastic situation if the players are against me. I could say we don't have results because the players are against me."
Mourinho made only three changes to the team that had lost 2-1 in a stormy game at West Ham United on Saturday, which left him facing a Football Association misconduct charge after being sent to the stands at half-time.
He was also recently fined £50,000 ($76,560, 69,370 euros) and given a suspended one-game stadium ban by the FA -- which he has appealed against -- for criticising the referee after the loss to Southampton.
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Chelsea made a bright start, but lost striker Diego Costa to a rib injury in the 33rd minute following an earlier collision with Butland.
Mourinho revealed afterwards that the Brazil-born Spain striker had been taken to hospital.
Stoke went ahead in the 52nd minute when Jon Walters slammed a shot in off the underside of the crossbar from the edge of the box.
But Chelsea refused to accept defeat and equalised in the 91st minute when Loic Remy, Costa's replacement, fired home from close range after Kurt Zouma had flicked on Willian's left-wing corner.
Stoke lost Phil Bardsley to a second yellow card moments later, but although Chelsea failed to make their numerical advantage count in extra time, Mourinho felt that his players had sent a message to their critics.
"What the players did tonight (Tuesday) is face some people that write and speak, and said: 'You are stupid'," he said.
"What some people write and say is really bad for the players. Maybe they think my players are like them when they were players. My players don't do that. They tried everything."
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