The Brazilian, 35, joined Chelsea from Paris Saint-Germain on a free transfer, signing a one-year deal with the option of a further year.
Silva's new manager Lampard himself stayed with the Blues until the age of 35, before spells with Manchester City - where he scored eight goals in 38 appearances - and finally New York City.
The defender insists the confidence Lampard showed in him, despite reaching the twilight years of his career, helped him decide to switch Paris for London.
"First and foremost, I came because Chelsea is an amazing team and one of the biggest clubs in the world," Silva told the club's official website.
"It helps when you’re welcomed so warmly, particularly by a coach who played until relatively recently, because in football there are very strong preconceptions regarding the age of players.
"But I know about his experience when he first arrived at Manchester City, theoretically at the end of his career. He endured the same sort of thing as I’ve gone through.
"So even from the outside, without knowing me personally, he knows how hard I work to remain at the highest level. The confidence that he showed in me - that was all it took.
"For me, that confidence was the single most important factor. To hear that he believed in me, that he thought that I could come here and make a positive contribution."
Silva has joined Chelsea in the midst of a mass recruitment drive, with Lampard bringing in the likes of Timo Werner, Hakim Ziyech, Kai Havertz and Ben Chilwell.
And having commenced training with his new team-mates, the centre-back is convinced his new side can reach the very top this season.
"With our young group, as well as some more experienced players like our captain [Cesar] Azpilicueta, we’ve got all it takes for a great season.
"You can be sure that I’ll do everything I can to repay that confidence out on the pitch. We’ve signed some top players and we’ve got a squad capable of putting together a great season so that’s what we’re going to try and do."