The Frenchman had a golden opportunity to grab a win for the Gunners late on in Saturday's game but could only fire an effort wide when unmarked in the penalty area.
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The former Montpellier man has moved to distance himself from comparisons with departed captain Robin van Persie, and claims he will not think about trying to replace the Dutchman.
"This could have been the dream start," Giroud told reporters. "But in Montpellier, I did not score for my first game, so I am not worried. Here, I will have plenty of chances and I will score.
"Last season was wonderful here for Robin, but before that, he took time to settle. I am just asking people to be a little patient with me.
"I am not here to make people forget Robin, because he has made beautiful things here, but I just want to bring something new.
"However, I am under no more pressure than that, because if I think about that when I go onto the field, I will play with the handbrake on.
"I know Arsenal is a great club and needs a striker ready and efficient quickly. I need games to get confidence, and it is up to me to succeed."
|33/1||Olivier Giroud is 33/1 to be the top Premier League goalscorer in 2012-13 with bet365|
"It is true that we do not always cross on the first chance, but it is something we have to work on," the 62-year-old said.
"This is important for us because teams come here to defend and we often got the possession. We need to be more effective in the final pass."
Giroud had to settle for a place on the bench on Saturday as Lukas Podolski started in a central striking role, and Wenger revealed that the German will regularly feature in the middle, rather than from wide areas, as many had predicted.
"I see him as a centre-forward, but he has some work to do, to change his game a little bit, make runs in behind to protect the ball and move around the box," the Frenchman added.
"When you come from a wide position, it is not easy to do that, but he will get there. He is a fantastic finisher, but had no real opportunity."