City’s players were not at their fluid best after manager Pep Guardiola rang the changes for the first game after the international break but they still did enough to earn a customary home win over the Clarets, who had lost 5-0 in their last four trips to Etihad Stadium in all competitions.
Silva put City on course for victory in the 12th minute, slamming home from close range after Phil Foden’s shot was saved by Nick Pope.
De Bruyne made sure of the victory in the 70th after pouncing on a loose ball in the area to sweep in a left-footed shot.
The visitors felt Riyad Mahrez fouled Ashley Westwood in the area as City attacked but nothing was given and De Bruyne thumped home unerringly.
“After the international break, it’s always a little bit difficult to come to a rhythm and I know how good Burnley are,” Guardiola said of a team that defended stoutly but was never likely to end its wait for a first win of the season.
“The team showed me the way we want to play and the shape we want to play in. All the guys gave everything, it doesn’t matter if the performance is good or not, it’s important that you fight for your teammates and everyone was incredible.”
City, which started the day in third place, stayed a point behind Liverpool, which beat Watford 5-0 earlier.
Raheem Sterling played the whole game as City’s central striker, two days after saying he would be open to leaving the club in the search for more playing time.
With first-choice City goalkeeper Ederson Moraes unavailable after returning late from international duty with Brazil, U.S. international Zack Steffen played his first Premier League match since January and his second ever.
Steffen played on Wednesday night in a 2-1 win over Costa Rica in Ohio, then took a charter to Frankfurt with U.S. Europe-based players to get back.