The Blues ended a run of four games in all competitions without a victory by beating Wigan 4-1 at Stamford Bridge on Saturday.
That result ensured Chelsea remained in third place after Tottenham's victory over Newcastle and Azpilicueta acknowledged that they must deal with the weight of expectation on their shoulders.
"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 pounds 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."