A lacklustre Madrid were soundly beaten 3-0 at Parc des Princes on Wednesday and have now won just two of their opening five matches in all competitions this term.
Navas, 32, made his second start for PSG after joining the Ligue 1 champions from Madrid in a deadline-day deal that saw Alphonse Areola head in the opposite direction.
Another former Madrid player, Angel Di Maria, scored twice for the hosts before Thomas Meunier added the final blow in second-half stoppage time.
"I am very happy because we won, not because we beat Real Madrid," Navas said.
"My team-mates put in a great effort, gave their best and we won, which is really important for us.
"I have no resentment towards anyone at Real. I just want to enjoy the moment I have here at my new club. Everyone is very motivated and I will defend my new colours as I did with Madrid.
"I want to continue writing my story. The one I lived in Madrid was very beautiful. I want to be happy here, enjoy every moment and for my family to be happy, that is the main thing."
Family, Navas claimed, was the reason he accepted PSG's offer of a four-year contract, and not because he had lost his starting place to Thibaut Courtois.
The Costa Rica international spent five seasons at the Santiago Bernabeu, winning the Champions League three times.
"I did not leave Real Madrid because of that," he said, when asked about the rivalry with Courtois.
"I left because my family and I thought this was the best option for us. I was welcomed as if I had been here for a very long time.
"The best I can give is to defend these colours to the death, to work hard and to try to win important games."