The Frenchman netted a first-half equaliser against West Ham before going on to set up Walcott later in the game, and maintained that he was always confident the goal would come
The Frenchman netted in the 41st minute to cancel out Mohamed Diame's opener, and went on to assist Theo Walcott's goal, Arsenal's second, with a fine through-ball before Santi Cazorla added a late third.
Giroud told ESPN: "I'm so happy tonight to score my first goal and to assist my friend Theo. I'm very happy, I feel relieved now and I hope so many goals come after.
"Everyone talked to me about this goal, everyone said: 'There's a goal coming soon, don't worry, we trust you,' and I keep confident, and I keep going, concentrate on the work in training and the game. So I keep confident."
Walcott hailed the Gunners' resolute defensive display, and hopes to have proven his worth to Arsene Wenger after starting most of the season on the Gunners' bench.
"It's always going to be a tough battle here," the England man added. "We haven't been here for a couple of years now. West Ham are always strong under Sam [Allardyce], and we showed good qualities, not just going forward, but defensively we were very strong.
"Maybe [we were] quite nervy in the first 10-15 minutes but the second half was much stronger. The true Arsenal, definitely.
"You've got to be patient in the game at times. I've always said I wanted to play, and that's the best way to answer your critics and hopefully now I can try and get a nice run in the team. The team is looking very strong and I'll be happy to be part of it."
The Gunners climbed up to fifth after their win at Upton Park, and the winger believes they have enough quality to sustain their push towards the top of the Premier League this term.
He continued: "We just [take it] step by step, every game as it comes, but we know where we want to be this season. We want to be hopefully right up there come the end of the season.
"We've got some great players like I said before that have come in and I think the way we came back today, I think we can definitely have a good push now."
Santi Cazorla once again played a starring role for Arsene Wenger's side, sealing the game with a superb third goal, that dipped and swerved past Jussi Jaaskelainen and into the corner.
Walcott enthused of the Spain international: "You can see he's come to the Premier League and he's taken it by storm.
"He's an absolutely fantastic player to play with in training every day - the same as these guys who've just come in as well - but Santi is the one that has stood out definitely, and today he's our conductor.
"He just takes everything, and his goal was absolutely fantastic. I don't even know what footed he is, to be honest."