It was City's first clean sheet in the Premier League this season as they move up to 10th position in the table.
Riyad Mahrez almost opened the scoring inside the first minute but his long range-effort went narrowly wide.
City came close to scoring a second after the hour mark when David Luiz deflected Joao Cancelo's cross just wide.
Arsenal are now without a victory in their last 29 Premier League away games against 'Big Six' opponents since a 2-0 win at City in January 2015.
"They were a difficult team to break down, really well drilled and made it difficult for us," Sterling said after the game.
"In the end we deserved the win. We kept it tight and compact and deserved the win," said Sterling.