Second-half goals for the 20-year-old - his first in almost two years - and Lukas Podolski made sure of at least second place in Group B after leaders Schalke beat Olympiacos.
The Gunners must now win in Athens in a fortnight to stand any chance of finishing top and so head into the last 16 as one of the seeds.
Wilshere knows they have to get better and told Sky Sports 4: "We started off sloppy. But we've still got one game left. We need to finish top.
"The first half was hard - we were sloppy, we gave ball away and weren't ourselves so needed to come and get an early goal."
He added: "We have to step it up.
"Tonight wasn't good enough, especially in the first half.
"We've got to play the best teams to win it so it doesn't matter who we get."
Wilshere ran to embrace the Arsenal physio after scoring and of his own recovery, he said: "It (fitness) will come - I've never been injured before so I can't tell you how long it will take."
Podolski echoed Wilshere's sentiments about the performance, saying: "In the first half we lost a lot of balls.
"The first goal from Jack was good for us.
"We were after three points and we are very happy about this."
Gunners coach Arsene Wenger also refused to consider winning the competition, telling Sky Sports News: "We don't worry too much about that.
"When you are in this competition you just want to get into the next round.
"Now we go into a cup competition."
Of Wilshere, the Frenchman admitted he had some similarities with former midfielder Cesc Fabregas.
"Yes. He is getting stronger physically and tonight he got in the box," he added.
"He is a complete midfielder - he can defend well and attack well.
"This sort of situation will give him confidence."
Of the news of the sacking of Roberto Di Matteo, whose departure came just months after he secured both the Champions league and FA Cup for Chelsea, Wenger added: "Yes, it is sad.
"I know how much heart a manager puts into his work.
"So it is very sad."