The visitors paid for two moments of dreadful defending as Shaun Wright-Phillips was gifted acres of space to fire Chelsea ahead in the first half before the same player pressurised Joleon Lescott into heading an own-goal winner in injury-time.
In between, Ayegbeni Yakubu had levelled for Everton with a fine finish after Mikel John Obi was handed a straight red for a lunge at Phil Neville.
"It was all down to hard work and team spirit - that's how we train and we had to take it on the pitch," Wright-Phillips told Sky Sports 1. "The character, team spirit and everything we showed today when down to 10 men - Everton are a hard team with good players.
"It's easy to lose games, we had to stay compact and concentrate."
The man of the match was confident enough to try to hijack Lescott's own-goal as his after he leapt with the Everton defender.
"Of course I touched it and I'm going to claim it."
"It's really hard to take," he told Sky Sports 1. "In the big games you've got to go right to the end, but I have to say Chelsea are a really good side.
"But we have still got a chance.
"It was a hard place to come and we are disappointed to lose the game but we're not out yet."
The Scot also questioned whether Lescott was fouled in the process of scoring his own goal.
"I just don't know if Shaun Wright-Phillips holds him down slightly," he observed.