Holland forward van Persie has joined Manchester United in a ?24million deal following a public statement earlier in the summer that the former Gunners skipper would not be signing a new deal.
Arsenal manager Arsene Wenger accepted his side lacked a cutting edge in attack following Saturday's goalless draw with Sunderland at Emirates Stadium.
Giroud, though, could have made the difference after coming off the bench in the second half - but when the big chance came as he ghosted into the penalty area with nine minutes left, the French striker rushed his shot and fired wide.
However, after his 21 goals helped Montpellier to the Ligue 1 title last season, the 25-year-old is confident things will all soon fall into place with Arsenal.
"It was a game ball, but the coach told me that I had not to worry about it," said Giroud, signed in what was understood to be a ?10million deal.
"This could have been the dream start, 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."
Giroud added: "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."
Wenger accepted Arsenal lacked balance in attack against Sunderland.
Giroud believes that is something which can easily be rectified.
"It is true that we do not always cross on the first chance, but it is something we have to work on," he said.
"This is important for us because teams come here to defend and we often got the possession. We need be more effective in the final pass."
German international Lukas Podolski had started Saturday's Premier League opener at Emirates Stadium to spearhead the attack following the departure of van Persie.
However, Podolski had few chances to net on his Arsenal debut before being replaced just after the hour by fellow summer signing Giroud.
The 27-year-old bagged 20 goals for club and country last season, and Wenger feels the Gunners will soon see the best of the former Cologne marksman.
"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," said Wenger.
"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."
Arsenal are set to see midfielder Alex Song also leave before the transfer window after a deal, understood to be ?15million, was agreed with Barcelona.
It is another player whom Wenger has schooled for many years deciding to take their career on elsewhere.
"We are left with that decision and now we have to face with it, to deal with it in a correct way," said the Arsenal manager, who could now move to bring in another midfielder, with long-time target Yann M'Vila still on the radar, as Rennes hold out for a ?17million fee, along with Real Madrid's Turkish midfielder Nuri Sahin, who is also interesting Liverpool.
"We make the players here, all who have left have made a name here after coming very young - [Cesc] Fabregas, [Gael] Clichy, [Emmanuel] Adebayor, [Thierry] Henry, [Samir] Nasri and Song.
"He came at 17, had a lot of work to do, but he did it."
Wenger added: "I do not think it is linked at all with winning something or not. It is difficult to make an overall conclusion about it. Every case is individual. Maybe one day I will explain everything.
"However, part of our job is to influence people's life in a positive way.
"You would prefer to influence it in a positive way for yourself, but it does not always happen."