David Moyes praised his team's spirit as the Blues recovered from a first-minute setback to draw 1-1 with Arsenal on Wednesday evening.
And he insisted his side could have had a potentially match-winning penalty in the second half.
Theo Walcott put the Gunners ahead after just 52 seconds with a precise clipped effort, but Marouane Fellaini's 27th-minute leveller grabbed Everton a point they more than deserved.
"I thought we played really well tonight," said Moyes after the full-time whistle.
"It was a tough game, Arsenal are a really good team and we ran them close.
"Great credit to the players for the character they showed to come back after going a goal down so early, and we deserved a draw for the way we played."
With the hosts engineering the best chances throughout an entertaining clash at Goodison, the Blues boss was also keen to hail the commitment of his side until the very end.
He said: "I thought we kept it going in the second half. The players played really well tonight - it was a real big effort.
"We didn't start from the first minute but I thought we played until the final whistle. We had two or three really good opportunities to score in the last five to ten minutes. A couple of crosses whizzed across goal and we chipped a few balls into the box."
One moment that Moyes believed could have changed the game saw Steven Pienaar brought down in the box by former teammate Mikel Arteta, only for appeals for a spot-kick to be waved away.
He said: "We should have had a penalty. I thought we played against a really good Arsenal team and a lot of games are won by decisions referees make or don't make.
"We got nothing tonight whatsoever and the players had to earn everything they were going to get because we weren't going to get a lot, that's for sure."