And the Frenchman contrived to miss the best opportunity of the game when he fired wide from 12 yards - a chance that Wenger said he expected his player to take.
Martin O'Neill praises 'terrific effort' by Sunderland
The Gunners boss also admitted it was an opportunity that, had he been in an Arsenal shirt, Van Persie might have taken "because many times he scored last year".
Former Montpellier striker Giroud was set up by fellow
new signing Santi Cazorla
and the Spanish midfielder enjoyed an encouraging debut.
Wenger said his performance showed that there was no excuse concerning the lack of time some players had spent at the club.
Instead he believed the speed of Arsenal's passing was the biggest hindrance.
"Cazorla played very well, he just arrived a week ago so [the overall performance] is not necessarily linked with [new arrivals]," Wenger said.
"Some players changed position, like Podolski, and some players were not completely fit yet. Our formation with three strikers didn't work today. We had no space, the quality of the balls we got was not always great and not quick enough therefore it was difficult for them. Maybe we needed one more creative player."
Wenger added: "It's one game and it's too early to draw definitive conclusions out of a game like that because we faced a specific problem from a team who played very deep."
Sunderland boss Martin O'Neill admitted that his side were happy with a point.
"It was a tough fixture for us. We haven't got all our players in place, and we are hoping to add to the squad very shortly," he told BBC Sport. "In the sunshine we got very tired and with Arsenal forcing the issue, we did well to get something out of the game.
"We have goals to get at the other end, and that is something we will work towards because it was obviously a problem last season but I want to commend the effort of the team. In the first half when we had more energy we were always dangerous on the break.
"Giroud's chance was a really good one and naturally we would have been really disappointed if we lost. But at least we are off the mark away from home against a naturally very talented team."
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