The Blues ended a run of four games in all competitions without a victory by beating Wigan 4-1 at Stamford Bridge on Saturday.
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"We always have pressure," he told reporters. "We know we must win every game.
"The confidence in the squad has dropped a little bit but even when they scored, instead of being scared and dropping back, we had a good reaction.
"We know we have to stay in the top four for the stability of the club."
Azpilicueta has firmly established himself at right-back since his £8 million move from Marseille and was involved in two of Chelsea's goals against the Latics.
That win capped off a week in which the 23-year-old had also made his debut for his country in Spain's 3-1 victory over Uruguay.
"It was the first time I have been involved in so many goals," he added. "It is a good thing to improve and create goals for the team."