The France international bagged his first league goal since a £12 million summer move from Montpellier in the 3-1 win at West Ham as the Gunners recovered from the disappointing home defeat by leaders Chelsea.
Although Arsenal, who had beaten Olympiacos in the Champions League earlier in the week, are currently just outside the top six and seven points off the pace, Giroud believes there is plenty more to come.
"We will keep going and stay confident for the future. I think we have a very interesting role in this championship," Giroud told Arsenal Player.
Giroud produced a neat close-range finish when turning in Lukas Podolski's cross to equalise at Upton Park, and is relieved to have got off the mark in the league.
Speaking to French broadcaster Canal+, Giroud added: "I expected it to be difficult at first because the Premier League is a level higher and I have had to adapt.
"It is nice to have scored my first goal, that will give me a lot of confidence for the future.
"I feel good about this team and I am confident that I will score more goals. Why not a dozen like in my first season at Montpellier?"
Podolski opened his Arsenal account in the 2-0 win at Liverpool before the last international break.
The German, signed from Cologne in an £11million deal, feels Giroud can now play with more freedom when the domestic action begins again with a trip to Norwich on October 20.
"Everybody wants goals from strikers and it was not easy for Olivier," Podolski told Arsenal Player.
"You must wait for your chance, keep fighting and the goals will come."
Arsenal could have Poland international goalkeeper Wojciech Szczesny back from an ankle injury for the trip to Carrow Road.
Vito Mannone has proved an able deputy in his seven appearances so far.
The 24-year-old spent some of last season on loan at Hull, but played down as "crazy" reports he was concerned over his long-term future with the Gunners.
Mannone, signed from Atalanta in 2005, told the club's official website, www.arsenal.com: "Neither myself nor anyone from my family or friends have spoken about any deal. At the moment I am really happy and just focusing on training and playing well.
"I am trying to be the number one for this club and win trophies, that is my only concern and I would like to make that clear.
"The boss has given me the chance. I have to keep going and prove myself in every single game.
"This is my best time at Arsenal so I would be crazy to not take the chance to put up a fight for my place.
"I am really happy and honoured to play for this club."
Arsenal will have nearly all of the first-team squad away on international duty.
Alex Oxlade-Chamberlain, Theo Walcott, who came off the bench to score at West Ham, and defender Kieran Gibbs have all been called up by England. However, the left-back is a doubt after he was substituted at Upton Park because of a thigh problem.
Meanwhile, midfielder Jack Wilshere continued his rehabilitation from injury with a 74 minute run-out and played a part in both the Gunners goals in their Under-21s 2-0 win over Reading on Monday evening. Emmanuel Frimpong, who has been linked with a loan move back to Wolves when fully fit following his knee ligament problems, played the full 90 minutes of the game.
Right-back Bacary Sagna featured also played an hour in the fixture after missing the end of last season and the European Championships because of a broken leg.