Chelsea began the season well under Lampard as they topped their Champions League group, but the club great was dismissed in January after a poor run of form and replaced by Thomas Tuchel.
There was a huge transformation under the new manager, the London side went on to win their second Champions League crown in the club history.
Silva, who was signed by Lampard from Paris St Germain, said the former boss could be proud of his contribution to their success.
"Without Lampard, I would not be here," Silva told French broadcaster RMC Sport. "I'm very happy, and I hope he will be too."
The Brazilian who was replaced by Andreas Christensen after getting injured in the first half, reflected on the win after falling at the final hurdle with PSG last season.
"This is the most important moment in my career. I'm truly very happy. We didn't manage it with PSG but today I'm happy. I hope PSG do it too," Silva said.
"Each time PSG were knocked out, it was made out to be my fault. It's a shame. I did everything I could for the team."
Kai Havertz, another of Lampard's signings, said he was delighted to have scored the winning goal to secure his first major trophy.
"I've waited 15 years for this moment and now it's here," Havertz told BT Sport.
Fellow midfielder Mason Mount, twice an FA Cup runner-up with Chelsea, said he had achieved a lifelong dream to win a title with his boyhood club.
"To go all the way in the Champions League, we played some tough teams. At this moment, we're the best team in the world. You can't take that away from us," Mount said.