Wenger's side ended a wretched run of four successive defeats as they won 2-0 at Milan in the Europa League last 16 first leg on Thursday.
With Arsenal having lost eight of their previous 13 matches in all competitions, Wenger was facing a fight to save his job ahead of the trip to Italy.
Arsenal are languishing 13 points adrift of the Premier League's top four and a dismal 3-0 defeat against Manchester City in the League Cup final exposed numerous flaws in Wenger's squad.
And ahead of Watford's visit to the Emirates Stadium in the Premier League on Sunday, Gunners midfielder Wilshere conceded he and his team-mates had been performing well below their best for several weeks.
Also Read
"We know we haven't been good enough. We are aware of it," he said.
After almost 22 years in charge at Arsenal, Wenger has presided over a major drop in standards in the last few years and Gunners fans have been increasingly vocal in their calls for his dismissal.
"If anyone tells you you haven't got to prove anything to anyone, they are wrong," he said.
"You are always trying to prove yourself to people and play well in the next game.
"We have got to use this win as a target. We have got to reach these levels every week.
"We have got another game on Sunday and then next week, the job in this competition is not done yet...this is AC Milan.
- Painful defeat -
==================
Like Wenger, England international Wilshere also faces an uncertain future with Arsenal.
The 26-year-old has little more than three months remaining on his current deal but negotiations over an extension remain at an impasse.
Wilshere started the season needing to prove his fitness and form in order to earn a new contract.
He has returned to being a first-team regular after been limited to Europa League action earlier in the campaign.
"All I can tell you is that we are no closer to sorting anything out," Wilshere said.
The Hornets have won three of their last four matches to consolidate their mid-table position and boss Javi Gracia is in confident mood.
"I think always is a good moment to play. I try to enjoy every game. Every game is a chance to get points and I don't think anything else," he said.
"Always it's complicated to play a big club or team, but we are confident we can do it.
Disclaimer: No Business Standard Journalist was involved in creation of this content