The Gunners missed the chance to put daylight between themselves, Tottenham and Chelsea in the race for Champions League qualification after the Toffees utilised all their physical qualities to keep alive slim hopes of gate-crashing the top four.
As well as several hefty challenges from the visitors - with midfielder Darron Gibson fortunate not to be sent off for twice pole-axing Theo Walcott in the first half - television pictures showed England international Wilshere, just back from a six-week injury lay-off, involved in a bust-up with Everton's Kevin Mirallas as the players walked off at the break.
He reacted angrily after appearing to be squirted with a water bottle - an incident which could yet be reviewed by the Football Association.
Despite the obvious frustrations, Wenger maintains his side still have what it takes to finish the campaign on a positive as they regroup for Saturday's trip to Fulham.
"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," Wenger said. "Let's look at ourselves, come back in the next game and win it. This still keeps us in a strong position in the race for the top four.
"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."