Wilshere started on the bench in Tuesday night's Champions League clash at home to Napoli as record signing Ozil scored his first Arsenal goal and set up Olivier Giroud to make it a 2-0 win for Arsene Wenger's side.
But the England midfielder sees competition for places at the Emirates as a positive thing.
Wilshere said: "It was just a formation thing. I am feeling fine, (Tomas) Rosicky is coming back from injury and needed a game.
"I have played a lot of games recently, so I was just happy to get on and help at the end.
"We still have Santi Cazorla to come back, so we are going to have a great squad."
Wilshere continued: "When I (first) broke into the team, it was similar - there was (Cesc) Fabregas, (Abou) Diaby, (Alex) Song, players like that.
"It brings the best out of me and it will bring the best out of the other players as well.
"If we can keep everyone fit, we are going to be real contenders in all four competitions.
"We have not really had this over the last few years, all of this competition for places.
"We are all excited and we just have to keep it up now."
Ozil is in no doubt that Arsenal, who sit top of the Premier League table, have the quality needed to last the distance in the race for trophies.
Ozil told Arsenal Player: "You can see in training how technically good this team is and that showed on the pitch too.
"The team supports me so well. We've got some great players in our ranks. They make life easier for me on the pitch and I am thankful to them for that.
"Our aim is to play attractive and successful football. We did that against Napoli.
"What is important for us is that we take the next step forward as a team. We want to keep on developing."