Chelsea were meant to have improved since finally ending their long wait for Champions League glory, but Cole acknowledged they showed none of the "fight, desire, passion" that saw them claim club football's ultimate prize last season.
He tweeted: "Wellll that was a lesson, absolute joke no fight, desire, passion, and a big reality check!!!!!! #sorrycfcfans."
The loss was one of Chelsea's heaviest since Roman Abramovich bought the club and would have put a real dampener on the Russian's celebrations after he won his long-running legal battle with Boris Berezovsky earlier in the day.
It also may have sowed the first seed of doubt as to whether the £80million he spent on his favourite hobby this summer was value for money.
It had certainly looked that way when Chelsea stormed to the top of the Barclays Premier League, but Friday was another failure with silverware on the line following last month's Community Shield defeat.
Gary Cahill, who scored Chelsea's consolation after Radamel Falcao's hat-trick and Joao Miranda had crushed them, told Sky Sports 2: "We fell to pieces. We started the game sloppy and that went right through the game. If we're honest with ourselves everyone was below par.
"It was embarrassing at times. We let ourselves down and passed up a great opportunity. Obviously we will look to bounce back in the league as today was unacceptable."
Chelsea defended against Falcao as if they were completely unaware he was one of the best finishers in the world, rather than a player with whom they had long been linked.
Man-of-the-match Falcao admitted there had been interest in him from England this summer. He added: "The club that made the effort to keep me was Atletico Madrid and, today, that was rewarded."