Lampard's Chelsea tenure began with a 1-1 draw against Bohemians in the Blues' opening pre-season game in Ireland on Thursday (AEST/NZST).
Chelsea's record goalscorer took over as boss on July 4, following an impressive debut season in coaching at the helm of Derby County.
And Azpilicueta has hailed the impact his former team-mate has had on Chelsea's squad, acknowledging that displays under former coach Maurizio Sarri were not always up to scratch.
"It is true that last season, for periods, we didn’t give the fans what they expected," Azpilicueta said. "Our performances were not good enough.
"We turned it around and won the Europa League but you can see that this season all of the fans are very happy we have the gaffer back at the Bridge.
"Now we have to give it back by working hard on the pitch, by winning games. We are very happy to have him with us. I can’t add anything to his career at Chelsea as a player.
"But, over his first week in training, I’ve seen the way he wants to play, the way he wants us to feel the shirt, the way he wants us to train every day, it is something special."
Azpilicueta expanded on his plaudits, adding that Lampard's commitment to Chelsea during his playing career sets the perfect example for the current players to follow.
"He was here for a very long and very successful period, and always set an example: how he wanted to break every record or improve himself in every session," Azpilicueta said.
"We have to be grateful and feel the passion for the shirt. We are very excited and the only way we can carry it on is by winning together, getting the feeling back of being all together as a club.
"We have to take responsibility and know that there are a lot of things [going on] around us, but we have to work hard to get better."
Lampard's first competitive game will be against Manchester United on the opening weekend of the Premier League season, before Chelsea faces Champions League winner Liverpool in the UEFA Super Cup on 15 August (AEST/NZST).