The full-back scored his first competitive goal for the club on Tuesday as the Blues recorded a 2-0 win over the Gunners in the fourth round.
Azpilicueta now hopes that Chelsea can leapfrog leaders Arsenal in the league standings when they travel to Newcastle United on Saturday.
"Arsenal now have two really hard games but we don't need to look at other teams," he is quoted as saying by Sky Sports News. "We need to keep working.
"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."
Chelsea are currently two points behind Arsenal in the Premier League table.