Theo Walcott gave the Gunners the lead inside a minute at Goodison Park, only for Marouane Fellaini to bring the hosts level, and the two sides had to settle for a point each.
However, after a summer of change at the Emirates Stadium and Wilshere's long-awaited comeback to the team, Wenger believes that improvement is the logical next step.
Speaking to reporters after the match, the Frenchman observed: "That's our job [pushing on] now, you know. We have built a new team, we now have Jack Wilshere back and hopefully we can go from strength to strength.
"But we have the spirit. We need some overall improvement in the final third but, overall, when you have the spirit, you have a chance. We have to show that now, we have to be consistent."
|STRENGTH TO SALE?|
|9/2||Theo Walcott is 9/2 with William Hill to leave Arsenal for Liverpool in the January transfer window|
Wenger also praised Arsenal's character and effort despite failing to hold on to their lead, noting that the team have had three matches in the last week, yet they still competed with the Toffees.
"Overall we had a good physical intensity," he argued. "We had a good start and after that I feel Everton came back in it. We have regrets over the goal we conceded but overall I believe that 1-1 is a fair result. It's a good point here and it was a game of remarkable intensity.
"We needed to show great spirit today to get away with a point because they were up for it and overall I can say I am only pleased with the effort we have put in the game.
"We played last week in the Champions League Wednesday night, Saturday at [Aston] Villa, today at Everton, we have to give a lot and we have some players who play on the edge and, hopefully, we recover for Saturday."
After a tough start to the season, Wenger is hopeful of a much easier period, with trips to the Manchester clubs, Liverpool and Everton all now out of the way.
"We hope so [to have an easier few games]," the Gunners boss continued. "When you look at the moment in the Premier League we have played 14 games, eight away now, we have played at Liverpool, at the two Manchester clubs away from home so of course if we play well and make the results at home, we have a chance to come back."Wenger also revealed that Walcott picked up a knock in the second half and that Laurent Koscielny could be out for three weeks with a calf injury.