Nigerian Victor Anichebe turned in a man of the match performance as the Blues earned a 2-2 draw at Tottenham on Sunday - a result which leaves them four points behind Arsenal, the side currently occupying the Premier League's sole Europa League spot.
It could have been better for David Moyes' men, who had led until Gylfi Sigurdsson stuck a close-range equaliser just three minutes from time.
And while Anichebe admits it was a blow to miss out on what would have been a precious victory, he believes the performance proved Everton are up for the fight with seven games to go.
"We're definitely frustrated," the striker says."You could see at the end how disappointed we were. We played so well and though we were pegged back at times, we showed our resilience and I thought on the counter we were unbelievable, and I felt like we could score at any time. So not to come away with the three points is hard to take but we're still in the race for the top five spots and hopefully we can continue playing well until the end of the season."
Anichebe himself had a golden chance to win it for the Blues in the dying minutes, only to be denied by a fine smothering save from goalkeeper Hugo Lloris.
"I didn't realise he was that quick," admitted the Academy product. "I was thinking, 'should I have taken it sooner'. I tried to chip it in the end and it wasn't meant to be.
"I'm happy though. I thought I had a good game. It would have been nice to round it off with a goal at the end but at least I was there to try to score and hopefully from now until the end of the season I can keep getting more chances, keep playing well - that's the most important thing - and hopefully get on the scoresheet too."
After falling behind to Emmanuel Adebayor's opener just 35 seconds in, Everton hit back through a Phil Jagielka header and Kevin Mirallas' second goal in successive games.
Moyes' men were joined in the capital by a strong travelling support and Anichebe added: "The fans were amazing today. There were times when we were dead on our feet out on the pitch but them singing was like a 12th man and really got us going again."