"We can't dwell on this result. We could be disappointed if we win against Chelsea and then lose at Wigan," said Walcott.
"All the fans, you could sense the atmosphere, the fans were right up for it, just like the players were and we just didn't let Chelsea play.
"We were ready and wanted to do it for our fans because in recent years we let them down in these big matches and we just need to build on that now."
England international Walcott was pleased to have secured a win over the Blues who have held sway over the Gunners in recent meetings.
"It was nice to see an Arsenal team actually do everything right and hopefully we will just continue to do that," he said.
"All 11 players defensively were pressing at the right moments and we made Chelsea look quite average at times, but they are such a strong team and will be right at the top come the end of the season."