The match turned on the introduction of Giroud as a 69th-minute substitute after a Jermain Defoe penalty had equalised Sanchez's first-half opener to give the home side a flicker of hope.
Wenger hopes Arsenal’s trophy hunt will keep Sanchez, Ozil
Giroud scored with his opening two touches, first with a left-foot shot from a Kieran Gibbs cross on 71 minutes and then with a near-post header from a corner five minutes later.
Sanchez scored Arsenal's third goal in seven minutes from close in after Gibbs's shot had come off the post.
Santi Cazorla eager to extend stay at Arsenal
Defeat means Sunderland have made the worst start to a season in Premier League history.
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