Czech Republic international was brought into the fray after Ross Turnbull's red card and saved from the spot, despite being rushed on without having a proper warm-up
Cech was brought on after Ross Turnbull had been sent off and immediately kept out Leighton Baines' penalty kick, as well as the resulting rebound, en route to a 2-1 victory.
Despite the red card Cech praised the performance of the former Middlesbrough man, who started as manager Andre Villas-Boas looked to give some of his key players a rest.
"We had agreed with the manager that he would rest me in this match because the schedule has been busy for us," Cech told Chelsea's official website.
"Ross was doing superbly until the sending off and deserves credit."
Cech said it was difficult being brought on without a full warm-up and revealed how he prepared to face Leighton Baines' spot kick.
"Ahead of every match I always do the same warming up routine whether I play or not, but after sitting on the bench for an hour you are a bit stiff," he said.
"It was very quick and I didn't basically have the time to warm up myself again. I went on the pitch straight away.
"He [Baines] already took one penalty against me. I was decided that I would be waiting until the last possible moment and it paid off.
"He aimed into the centre and I punched it, however, straight back to him. But I coped with the rebound also by jumping at his feet and then [Tim] Cahill headed wide.
The 29-year old had to make further saves and admitted that it was not a vintage Chelsea performance, but was impressed with the team's ability to battle for a win in difficult circumstances.
"Shortly after the penalty one of the Everton players headed on the far post and I had to make a big jump to get to it," he said.
"Then I had to save against a dangerous shot in the final minute.
"We made the best out of the chance eventually. But even while they had a one-man advantage we were playing well and we are glad to have grinded it out."
How do you stay up with football when on the move? With http://m.goal.com –your best source for mobile coverage of the beautiful game.