The Gunners ended the campaign in strong form, recovering from defeat at Tottenham in March to beat Bayern Munich away and then win eight from the last 10 league games as they battled back to clinch fourth spot ahead of their north-London rivals - and with it another shot at Champions League football.
By contrast, Arsenal had secured only four victories from the opening 10 matches - which included defeats at Chelsea, Norwich and Manchester United. Indeed, Arsene Wenger's side dropped as low as 10th place when they were beaten at home by Swansea on December 1.
Wilshere, 21, knows it is a scenario which cannot be repeated if Arsenal are serious about lasting the distance in 2013/14.
"The games earlier on don't decide our season but we have got to think that we have to win them, because that is what champions do," Wilshere, who is set for ankle surgery which will rule him out of England's friendlies against Republic of Ireland and Brazil, said in an interview with Arsenal Player.
"The likes of Manchester United go away and win games and leave you thinking, 'they keep winning and winning' and at the end of the season the points tally shows that.
"That is what we have to do [to challenge for the title]."
Wilshere feels Arsenal must step up their form in the head-to-head against their direct rivals, having lost both at home and away to Chelsea, been beaten at the Emirates Stadium by Manchester City and having drawn with United when they came to north London just after being crowned champions.
"We also have to improve our record against the big teams," the England man said.
"It has not been great this season and if you want to become champions you have to beat those sides."
To help facilitate closing the gap on United, City and Chelsea, Arsenal are expected to significantly bolster their squad this summer as Wenger finally dips into his £100million war chest, set to be backed up by new lucrative marketing deals.
The club are also set to get Andrey Arshavin, Marouane Chamakh, Nicklas Bendtner, Denilson, Sebastien Squillaci and Korean forward Park Chu-young all off the wage bill.
However, Gremio have suggested they are unlikely to take up a permanent deal for Andre Santos, who is currently on loan back in Brazil.
France Under-21 forward Yaya Sanogo, on a free from Auxerre, will be the first of several new arrivals.
While Wenger is understood to have targeted offensive additions this summer, Arsenal are set to face stiff competition for prime targets - with Fiorentina's Stevan Jovetic and Real Madrid frontman Gonzalo Higuain both also of interest to Juventus.
The future of captain Thomas Vermaelen, meanwhile, is uncertain after the Belgian defender found himself relegated to the bench over the closing weeks of the season, and could now look to secure first-team football elsewhere to protect his World Cup dream.
Full-back Bacary Sagna has yet to agree a contract extension and has been linked with Paris St Germain.
Swansea's Ashley Williams and Borussia Dortmund defender Lukasz Piszczek are two reported targets should Wenger decide to overhaul the rearguard again in the summer.
Arsenal has also been suggested as a possible destination for Brazil goalkeeper Julio Cesar, who is set to leave relegated QPR.
Gunners stopper Lukasz Fabianski, though, feels he has shown more than enough quality to challenge fellow Pole Wojciech Szczesny for the number one jersey, having impressed when recalled to the side with consecutive clean sheets in the wins over Bayern Munich and Swansea.
The 28-year-old said to the club's official website, www.arsenal.com: "It is just about focus, being vocal, being ready, being switched on and stuff like that. I know that with my abilities, I will be good.
"Maybe that's the way the boss sees it now, that I went through a total mental transformation.
"I know what I am capable of. I know my strengths, I know my weaknesses and I know how to deal with it so that's the main thing for me."