The Brazilian centre-back was speaking after the Blues' 1-0 win over Manchester City in Saturday's Champions League final at Estadio do Dragao.
Silva became the oldest player to appear for Chelsea in a major European final, overtaking Claude Makelele, but the 36-year-old lasted only 39 minutes before limping off injured.
Nice to hear Edouard Mendy and Thiago Silva thank Frank Lampard, too. Lampard, like Vilas-Boas, fired the season Chelsea win the Champions League.— Darren Tulett (@1DarrenTulett) May 29, 2021
He is also only the fifth player to play in the Champions League final in consecutive seasons with different teams, after Marcel Desailly, Paulo Sousa, Samuel Eto'o and Alvaro Morata.
Kai Havertz netted the only goal of the game in Porto three minutes after Silva exited the field as Chelsea became the third English side, after Manchester United and Liverpool, to lift the trophy more than once.
The victory was particularly sweet for Silva, who was a beaten finalist with PSG against Bayern Munich last August in his final game for the club before departing as a free agent.
The French giants fell short in all eight of Silva's seasons at the Parc des Princes, but the experienced defender feels he was unfairly singled out for blame after each disappointment.
"This is the most important moment of my career," he told RMC Sport. "It's unforgettable. I didn't manage it with PSG but today I'm happy. I hope PSG do it one day, too.
"Every time PSG were eliminated, people tried to find a culprit and it was always me. That's a shame because I really gave everything.
"But I hope they win it because I have left a lot of friends there. I will always stay red and blue."
Silva arrived at Chelsea shortly after PSG's loss to Bayern last August after being persuaded to join by then-head coach Lampard.
The former midfielder guided Chelsea through the group stage unbeaten, but he was sacked in January and replaced by Tuchel following a disappointing run of domestic results.
But while Tuchel – the third different German coach to win the Champions League in the past three seasons – has received much of the plaudits, Silva was also keen to praise Lampard.
"This is special, in what is my first year here," he said. "Tuchel has changed the team mentality, but it is also important to talk about Lampard.
"Without him I would not be here. So, thanks to him for letting me sign here. I'm very happy and I hope he is, too."
Including his 39-minute appearance in the final, Silva featured eight times for Chelsea in their successful Champions League campaign.
The former Milan man, who won all four of his duels against City before leaving the pitch, is set to be out of contract again next month and has yet to agree fresh terms.
Asked if he will stay on at Stamford Bridge, Silva said: "We will see. We must take it season by season."