The Spanish full-back netted the opening goal of the game - his first ever goal for Chelsea - as Jose Mourinho stretched his unbeaten record against Arsene Wenger to nine matches.
Arsenal face a difficult run of games which sees them face Liverpool and Manchester United in their next two league games, but Azpilicueta says Chelsea are just trying to concentrate on their own performances.
"Arsenal now have two really hard games but we don't need to look at other teams. We need to keep working," said the former Osasuna player.
"We are improving in every game and this is the most important thing. We are growing up as a team. We are in second, two points less than Arsenal, and we're now going to try and get up to first.
"We play in a difficult stadium as well in Newcastle on Saturday. It's always hard there. This league is really competitive."
Azpilicueta, who signed for Chelsea from Marseille in 2012, explained Chelsea's performance against Arsenal has lifted the entire squad.
"We felt really confident on the pitch and did a very good game," he said.
"We controlled most of the game. We had two or three more chances but the feeling inside of us was that we were dominating Arsenal in their stadium and they are top of the table, so it's very nice."