Cole, whose miss from the spot allowed Phil Neville to win last Saturday's FA Cup fourth-round replay, began the process of redemption in Tuesday night's Champions League last-16 first-leg win at FC Copenhagen.
But with the Barclays Premier League title all but surrendered, nothing less than fulfilling Chelsea's long-held dream of lifting the European Cup will do for Cole this season.
"It is something not many players from here have won so it means a lot to us," said Cole, a runner-up in 2006 with Arsenal and two years later with Chelsea.
"I have been in the final twice and defeated twice and it is not a good feeling.
"Hopefully this season will be different, it will be one to remember.
"I am just desperate to win the Champions League and if I can do it in a Chelsea shirt that will be an amazing achievement for me.
"After starting the season so well, we then had this dip in form, it wasn't going for us, and of course it has left us where we are now.
"It is not a good position for us, we know we have to do better, we want to do better.
"We are out of the FA Cup, the Carling Cup, we're never going to give up on the league but the Champions League is probably now the main focus for us to win because obviously not many people here have won that one."
Of his penalty against Everton, which he blazed high over the crossbar, England left-back Cole told Chelsea TV: "It's one of those things.
"Every day in training I take penalties and I score. I didn't mean to go that high obviously. I thought the goalie was going to dive that way so I tried to put it above him.
"It was not a good moment for me but I just have to keep going now and try and make it up for the team."
