Silva smashed the ball home via the crossbar past Bettinelli in Manchester City's 3-0 victory over Fulham on Saturday as he featured in his milestone game.
Speaking after the match, manager Pep Guardiola said: "To play 350 games, you have to be here for a long time, healthy, positive and with a good mentality.
"It is a lot of games so he has to be so proud. I think the fans are proud of him but he has to be proud of himself also. To do that in England, to survive in a physical league with his type of play, it shows his intelligence on the pitch. It is a big number."
The Spaniard has become a real fan favourite at the Etihad, after helping the 'Citizens' lift seven major trophies in the Sheikh Mansour era.
David Silva has scored 63 goals and assisted 115 in his 350 appearances across all competitions with City.