The Gunners moved into the fourth round of the FA Cup with a 2-0 win over north London rivals Tottenham at the Emirates Stadium.
Arsene Wenger's men now have a full week to recover before they resume Barclays Premier League action at Aston Villa on Monday January 13, when they will be out to maintain their one-point lead at the top of the table, with rivals Chelsea and Manchester City both in action earlier over the weekend.
Wilshere, who turned 22 on New Year's Day, is confident Arsenal have developed the mindset needed to stay the distance through until May.
"Something clicked towards the end of last season when we knew we had to get fourth, there were games we weren't playing well in, but we were still winning and that has carried on this season," Wilshere said.
"Sometimes you can't always play your best and you have to still win games - you have seen the likes of Manchester United and Chelsea do that in the past and we are doing it now."
Wilshere added: "Every team is going to go through a little blip. We had ours and luckily came back strong.
"We have won a few games in a row now. We are in a good place in the league, in the next round of the FA Cup and we have a break to recover."
Wilshere insists Arsenal will not listen to those who continue to question their character.
"We know what we can do," he said.
"There is great character in this squad, a great team spirit.
"We are genuine friends off the pitch. We have a mix of cultures - German, English, Spanish - but we all get on.
"That is helping us and we are concentrating on ourselves rather than anyone else."
There was a controversial end to Saturday's third-round contest, when Arsenal forward Theo Walcott made a "2-0" gesture to the travelling supporters behind the goal in the Clock End after missiles were thrown at the stretcher which was carrying the injured player around towards the tunnel.
The Football Association will wait until receiving the match report of referee Mark Clattenburg before deciding whether the incident warrants investigation.
Last month Wilshere was given a retrospective two-match ban for making an offensive hand gesture towards the Manchester City supporters which saw him miss the matches against Chelsea and at West Ham, on Boxing Day.
The England midfielder hopes his team-mate does not suffer the same disciplinary action.
"I think they (Spurs fans) were giving him a bit of stick. He's a bit smarter than me, I suppose," Wilshere said.
"It is football. They were giving it to him and got it back. People have to look at it as a bit of banter."
There were some ugly scenes in the away section, with coins and plastic bottles both seen to be thrown towards the stretcher - some of which hit medical staff.
Otherwise, however, the match had been played out in a terrific atmosphere, with both sets of supporters playing their part to lift the teams.
Wenger believes the Arsenal players have noticed the difference in feedback from the terraces this season.
"Yes I feel it and what is happening is there is a connection again between the team and the fans," said the Gunners boss.
"They look to be on the same wavelength and that was not the case last season and to get that back for us is very important - to have that strength at home will be vital until the end of the season."
Wenger accepts it is not the ideal time to be shopping for the required calibre of player to boost the Arsenal squad, but should Walcott's injury see their attacking options further depleted, then there could be little option but to strengthen this month.
"We are like everyone, we look out there and see what is happening, if an opportunity turns up," said Wenger, already without Nicklas Bendtner for at least a month because of an ankle problem.
"I said many times we don't rule anything out.
"But you know as well that all the big players are at big clubs at the moment and they go for important targets.
"Unless you have a club in a desperate financial situation it will be difficult."