The Frenchman, who signed for £12 million from Montpellier in the summer, had a golden chance to earn his side three points on his Premier League debut but fired wide when one-on-one with Black Cats keeper Simon Mignolet.
Wenger admits that a lot is expected of the 25-year-old, but feels optimistic that he can score goals in England as he becomes accustomed to the league.
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He told Sky Sports: "That's the burden you face when you take over from Robin van Persie because people are used to seeing him score.
"But he missed some chances as well and I feel Giroud will score goals but he rushed his chance a bit he had more time than he thought he had and I believe in time he will get there. I'm not worried too much about that and I'm sure I'm right."
Wenger hopes that the same will be true for all of his new faces, including Lukas Podolski and Santi Cazorla, whom he feels had a strong outing against Sunderland.
However, the Gunners boss was not satisfied with his team's overall performance against the Black Cats and accepts that they will need to improve in the coming weeks.
"They are good players once they are on the pitch they play. We have seen Cazorla today, it doesn't take him six months to adjust to play," Wenger continued.
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"Sometimes it's a little bit of an easy excuse. I believe our pace in our passing was too slow out from the back - it wasn't sharp enough, we usually do better on that front.
"It was frustrating because we played with the right attitude and the right focus but overall I believe our creative level was not good enough today for different reasons.
"Some was the speed of our passing, some to the quality of our movement but our formation wasn't right - as well, the three strikers didn't work.
"Another reason is that [Sunderland] defended very well, very deep and very focused highly on defending. As long as you don't score the first goal it’s a different game."