The Frenchman said: "When I came here the one thing I wanted to do was win. Last year was not easy because we had too many injuries and this year everything has changed.
The Frenchman believes the fightback will enable Arsenal to lay down a marker to their title rivals.
He said: "We showed everybody we can fight. Maybe something will happen this year. Maybe we can win something, I don't know. The squad are very compact. They want to win. They want to fight. They want to go back home and enjoy it. If you want to have that, you have to work very hard on the pitch.
"Last season, we drew our first game and that happened a lot last season. This year, we take three points. It was very important to win this game because we don't play in the Premier League on Wednesday which means we will have three points less."
In a public show of unity, Gallas called the players together for a post-match huddle seconds after the final whistle.
He explained: "We don't have to show everybody we're together, only for Arsenal.
"In pre-season we spoke a lot about that. We have to be compact, we have to play together. If we win, we win together. If we lose, we lose together.
"Today, I feel we were a little bit nervous. I know game by game we will play better, score a lot of goals and not [have to] fight how we fought today. I know if we play our football, everything will be okay."
The game was his first since being appointed captain by Arsene Wenger on Thursday.
Gallas said: "When he said 'you will be captain' I was very proud and said 'thank you'. I didn't think about being captain but, trust me, I'm happy.
"I'm 30 years old. I have experience and I have to speak with the young players to give them good advice.
"I don't have to shout because they know what they have to do. I have only to speak with them, that's it. For me the most important thing is the squad, it is not to be captain. I'm very proud to be captain of Arsenal because before me you had [Patrick] Vieira and Tony Adams."