A week after they failed to win at Fulham, Arsenal could again manage just one point as Sean Dyche's side held on for a goalless draw at the Emirates on Saturday.
Mikel Arteta's side remain six points behind Liverpool having played a game more, with the 10-man Reds twice coming from behind in a 2-2 draw at Anfield.
The Gunners, Premier League runners-up for the past two seasons, are also a point behind Chelsea, who host Brentford at Stamford Bridge on Sunday.
Arsenal enjoyed 76 percent of possession and a total of 13 shots, but could not find the breakthrough against an Everton side that defended resolutely and had only two shots in the entire game.
"We're very disappointed not to win the game because if there was a team obviously that deserved to win it that was Arsenal," said Arteta.
"We did everything that was necessary overall apart from the fact that we didn't score a goal. We conceded absolutely nothing, dominated all the direct play, didn't allow them to run and conceded no set pieces.
"But we go there in so many occasions, we created three or four big chances, didn't score from there... at the end you want to be rewarded for what you've done and the difficulty is that we haven't tonight.
"It's all the moments that we need to have a little more composure and more quality in those moments, that’s normal. That's it.
"Tracking back? Unbelievable. Putting in the high press? Unbelievable. Dominating a lot of things? Unbelievable, but the disappointing thing is that you want to come away winning after what you've done on the pitch."
