Blues right-back Azpilicueta grabbed his first goal for the club since arriving from Marseille 14 months ago when he poked the ball past Lukasz Fabianski in the first half, while a brilliant strike from Mata after the break condemned the Gunners to defeat in their fourth round clash.
"We tried our best as it was a difficult game because Arsenal are playing really well in the Premier League and Champions League," Mata told Sky Sports.
"They will always create chances with Cazorla, Ozil or Ramsey so it was an important goal to make more distance in the result.
"But today I'm more happy for Cesar, to be honest. He trains more than anyone and he fully deserves the goal and the celebration more than anything."
Chelsea's win came barely 48 hours after their victory over Manchester City in the Premier League, and Mata was proud of the displays from the two very different sides put out by Jose Mourinho.
"Looking at the opponents we had this week, they are probably two of the best teams in Europe right now and we beat them," he said. "So I think we are ready to try for the title, it's still soon in the season but we will try."
Mata, one of 10 changes made by Mourinho, made the most of his chance to impress his manager, after being frequently left out of the Chelsea side so far this season.
"I just try to play, and enjoy my football," added Mata.
"The manager is the one who makes the decisions, but I always try my best for this club and that's what I'll always do. Today I enjoyed it, and I felt proud for the fans, but I'm really happy for Cesar."
Mourinho opted out of the post-match press conference, leaving assistant coach Steve Holland to reflect on how the Chelsea squad has coped with a busy few days for the Stamford Bridge club.
"I think it's been a satisfying three days," said Holland. "You're not just talking about a team. The objective at Chelsea is always to compete in all four competitions.
"Given we had to play the game tonight, which I know Jose has documented was far from ideal with one day between the matches, it gave us no choice really but to use the full squad.
"Every player in the squad deserves great credit for their performances in the three days."