However, Moyes now faces the prospect of heading to the Etihad Stadium to face Manchester City at the weekend without his most consistent player.
"Leighton's got a hamstring injury. He felt it in the game and played on in the last 10 or 15 minutes," said the Scot.
"I felt we put in a lot of energy (in the game) but it might take its toll.
"We've got Gibbo (Darron Gibson) and Hibbo (Tony Hibbert) back and that has eased it a little bit but I am worried about Nikica Jelavic (ill), who has not trained since Saturday and on Tuesday I was not playing him."
Theo Walcott put Arsenal ahead after just 52 seconds but Marouane Fellaini's seventh Premier League goal of the season drew them level on the half-hour before former Toffees midfielder Mikel Arteta appeared to bring down Steven Pienaar, only for referee Mike Jones to award a corner.
"It was a penalty. We played Arsenal today, who are a really good team, but we also had very few decisions go for us," he said.
"Sometimes referees' decisions change the games. In some games something might go in your favour but we were never going to get that tonight.
"We got nothing whatsoever and anything we were going to get we had to earn on our own because we were not going to get a jot.
"There was an incident when the ball got chipped into the box later on and he gave a push on Fellaini.
"I've had a chance to see it and it was never anything like it. I found that one as mystifying as any really.
"We found it a difficult night from that point of view."
Arsenal manager Arsene Wenger was pleased to leave Goodison Park with a point despite his side dominating the pair's last 10 league meetings, winning eight and drawing just two.
"It was a game of big physical intensity. We had a good start and after that I felt Everton came back into it and their direct game was always a threat," he said.
"There are regrets on the goal we conceded but 1-1 is a fair result and it is a good point here.
"We needed to show great spirit to get away with a point and I am pleased with the effort."
Arsenal lost Laurent Koscielny to a groin strain after just four minutes and Wenger said he expected the centre-back to be out for three weeks.