The Gunners had won four league games in succession to move above both Tottenham and Chelsea in the race for Champions League qualification.
However, despite plenty of endeavour, Wenger's men were unable to breakdown a determined Everton, who utilised all their physical qualities if not collecting the win they probably needed to harbour any real hopes of getting into the top four.
Arsenal's best chances fell to Olivier Giroud, but the France striker failed to test Tim Howard and the hosts had to be content with a point which sees them move two clear of their London rivals below them, but having played more matches.
"We should have won the game, but it still keeps us in a strong position in the race for the top four," said the Arsenal manager.
"We had the chances to score, and that's my regret. Maybe we used power instead of technique too much in our finishing.
"My belief was always strong and still is. I am confident we will do it. The best way is to put the same effort into the next game."
Wenger also praised Everton for pushing the Gunners all the way in the game.
"You have to respect the effort Everton put in. They decided to make it very physical for us to disturb our game and sometimes went a little bit over the edge, but the referee had to make the right decision.
"I don't blame Everton for that. 'Protecting' is a big word, but they (referees) have to make the right decisions."
Wenger challenged his men to bounce back with a win at Fulham on Saturday.
"We are on a very strong run with a fantastic run so we do not want to complain too much about what didn't go for us," he said.
"Let's look at ourselves, come back in the next game and win it."
Everton boss David Moyes defended his side's approach.
He said: "Do you mean when we were a little bit rough with the tackling? Up north we do that quite often, that's actually allowed in football.
"We weren't going to come here and let Arsenal stroke the ball around and make 600 tackles."
Moyes added: "I am pleased with the performance, and with the result. You are never disappointed taking a point at the Emirates, you can't be with the quality they have here.
"We went to try and win it. We knew if we got three points here we would have a real chance (of the top four), but a point keeps us ticking along."
Everton remain in sixth place, a point behind Tottenham, but having played a game more.
Moyes accepts it will be an uphill battle to qualify for the Champions League.
"I think that result suits Spurs more than anybody... but I don't think we are out of place where we are.
"It is hard for us compared to clubs like Arsenal, but we go out on the pitch and do everything we can to beat them, make sure they don't beat us.
"Everton, as a team, played really well tonight [Wednesday].
"I would take it (the Europa League), but I am going to try and get the Champions League - if we are still getting that mention in mid-April, then it is not bad for Everton."